transactions of theamerican mathematical societyVolume 316, Number 1, November 1989
♦ DIFFERENTIAL IDENTITIES
OF PRIME RINGS WITH INVOLUTION
CHEN-LIAN CHUANG
Abstract. Main Theorem. Let R be a prime ring with involution * . Suppose
that 4>(xA,(x A)*) = 0 isa »-differential identity for R, where A; are distinct
regular words of derivations in a basis M with respect to a linear order < on
M. Then (j>(Zij,z*.) = 0 isa »-generalized identity for R, where z¡¡ are
distinct indeterminates.
Along with the Main Theorem above, we also prove the following:
Proposition 1. Suppose that * is of the second kind and that C is infinite. Then
R is special.
Proposition 2. Suppose that Sw(V) Ç R C LW(V). Then Q, the two-sidedquotient ring of R , is equal to Lw(V) ■
Proposition 3 (Density theorem). Suppose that DV and WD are dual spaces
with respect to the nondegenerate bilinear form ( , ). Let vx,..., vs, v [,...,
v's € V and ux,...,ut, u\,...,u\ G W be such that {vx,...,vs} is D-
independent in V and {«],...,«,} is D-independent in W. Then there exists
a e Sw(V) such that v¡a = v't (i = 1.s) and a*u¡ = u'. (j = l,...,t) if
and only if (v'.,Uj) = (v¡, «') for i = l,...,j and j = \,...,t.
Proposition 4. Suppose that R is a prime ring with involution * and that f
is a »-generalized polynomial. If f vanishes on a nonzero ideal of R , than f
vanishes on Q , the two-sided quotient ring of R .
The objective of this paper is to prove the following generalization of Khar-
chenko's theorem on differential identities to prime rings with involution (the
notation here is explained in §1 below):
Main Theorem. Let R be a prime ring with involution *. Suppose that
cf)(x¡ ', (x/)*) = 0 is a ^-differential identity for R, where A are distinct reg-
ular words of derivations in a basis M with respect to a linear order < on M.
Then cf>(z¡j,z*j) = 0 is a ^-generalized identity for R, where z(.. are distinct
indeterminates.
Received by the editors April 18, 1988.
1980 Mathematics Subject Classification (1985 Revision). Primary 16A38; Secondary 16A28,16A72, 16A12, 16A48.
Key words and phrases. Differential identity, generalized (polynomial) identity, left (Martindale)
quotient ring, two-sided (Martindale) quotient ring, prime rings with involution.
©1989 American Mathematical Society
0002-9947/89 $1.00+ $.25 per page
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252 CHEN-LIAN CHUANG
We have the following immediate
Corollary. Let R be a prime ring with involution *. Then any ^-differential
identity of R is a consequence of the basic identities (l)-(9) of §1 and *-
generalized identities of R.
There is a similar work [5] by Lanski. However the main result (Theorem
7) of [5] is false. Before proceeding, it seems proper to give the following
counterexample to Theorem 7 [5] (and Theorem 4 [5] as well):
Counterexample. (Using Lanski's type of notation [5].) Let Q be the field of
rational numbers and let F = Q(t), the field of rational functions in t with
coefficients in Q. Set 7? = Fn, the ring of all nxn matrices over F, and endow
7? with the transpose involution * ; that is, for x — (a.) E R, x* = (a..).
Define the derivation S on R by setting, for x = (a..) E R, x = ((d/dt)au),
where d/dt is the usual differentiation derivative with respect to /. Observe
that (x*)s = (xs)* for xeR.
Let etj (i,j = 1, ... ,n) be the matrix units of R ; that is, let et, be the
matrix of R with 1 in the (i,j) entry and zero elsewhere. Let
(I 0\
b = tien =i=i
2
VO nj
Define the derivation d on 7? by setting x - x +[b,x] for x e R. Now
consider s = xd + (x*) - [2b,x*] for x E R. We compute s — x +(x*) -
[2b ,x*] = xô + [b ,x] + (x*)s + [b ,x*]-2[b ,x*] = xs + [b ,x] + (xsf -[b ,x*] =
x +(x )* + [b,x—x*]. So 5 is always symmetric and hence exlse2x—ex2sexx =
0 . We have thus shown that
f(x,y) = ex|(xd + yd - [2b,y])e2x - eX2(xd + / - [2b,y])ex,
isa G*-DIof 7? in the sense of [5]. The derivation d is obviously outer and can
be chosen to be the first element in the well-ordering of the basis of the space
of derivations on 7? modulo the space of inner derivations (that is, M\M0
in [5]). Let 0 denote the empty set. Theorem 7 [5] asserts that f0(x,y)
= eX2[2b,y]exl - exx[2b,y]e2x is a G*-PI of 7?. But, since f0(ex2,e*x2) =
f0(eX2 ,e2x) = 2e2x jí 0, this is obviously false!
The crucial difference between Lanski's Theorem 7 [5] and our Main Theorem
above is as follows: In [5], * remains inside of derivation words. By introducing
d* for each derivation d on 7?, we are able to push * outside of the derivation
words in our Main Theorem and this gives the right form for doing induction.
Ignoring the falsity of [5], our improvement is to remove the multilinearity
assumption of [5]. Viewing the complexity of Kharchenko's analogous work
[4] for rings without involution, this part is quite intricate. For this work, a
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»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 253
structure theory for *-primitive rings with socle is required and this is developed
in §11. There, as auxiliary results, we obtain an interesting characterization of
the Martindale two-sided quotient ring and we also prove the useful *-version
of the Jacobson density theorem.
One minor difference worth emphasizing is that the coefficients of our *-
differential identities must be allowed to lie in the left Martindale quotient ring
of R, instead of the two-sided Martindale quotient ring as in Lanski [5]. This
is not simply a matter of generality and is actually crucial to our proof, as will
be seen in §IV.
Our method is by mixing Kharchenko's techniques with a theorem due to
Rowen. The material is organized as follows: In §1, we generalize Kharchenko's
original theorem slightly and meanwhile explain our notations. In §11, we treat
the case when * is of the second kind and introduce the notion of special
differential identities. In §111, we generalize those well-known results on rings
with nonzero socle to their *-versions. The proof of the Main Theorem is given
in §IV.
Along with proving the Main Theorem above, the interesting auxiliary results
obtained are singled out under the title "Proposition".
Most of the notation here is adopted from Kharchenko's original work [3, 4]
instead of Lanski [5].
I. Preliminaries
Suppose that 7? is a prime ring and that F is the set of all its nonzero
ideals. Let RF be the ring of left quotients of 7? relative to F ; that is, 7?^ =
lim Hom(Ä7,7?). Consider the subring Q of RF consisting of those elements
a E RF for which there exists a nonzero ideal Ia E F such that ala Ç 7?. Q is
called the two-sided quotient ring of 7? relative to F . The center of Q, denoted
by C, coincides with the center of RF and is called the extended centroid of
R.
Let Der Q consist of all derivations on Q. For o E Der Q and a e C,
define aa by xaa = (xa)a for x E Q. For a,p. E T>erQ, define x -
(xa Y - (xM)a for x E Q. o a and [o ,p] thus defined are also derivations
of Q. In this manner, Der Q is a right vector space over C and is also a
Lie algebra (over the subfield consisting of a E C such that a" — 0 for all
a E Der Q) . Der Q is called the differential Lie C-algebra of derivations in Q .
Let Der 7? consist of all derivations on R and let D = (Der 7?) • C. Obviously
D C Der Q .
By a differential polynomial, we mean a generalized polynomial involving
noncommutative indeterminates which are acted by derivations of 7? as unary
operations. We allow the coefficients of a differential polynomial to lie in RF .
Obviously, every differential polynomial can be written in the form cf>(xjJ),
where <p(zi..) is an ordinary generalized polynomial in z( and A are words
of derivations of 7?. <f> = 0 is said to be a differential identity for R if
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254 CHEN-LIAN CHUANG
cfi assumes the constant value 0 for any assignment of values from R to its
indeterminates.
The following basic differential identities hold in any prime rings:
( 1 ) (xy)a = xay + xya , where o E Der Q.
(2) (x + y)a — xa + ya , where o E Der Q.
(3) xa = xa - ax, where a is the inner derivation defined by a E Q.
(4) x[<T""' = (x")ft - (xß)° , where a,pE DerQ and [o,p] is their commu-
tator.p-times
,-*-,
(5) (■ ■ • ((x ")") ■■■)" = x , where o e DerQ and where p is the char-
acteristic of the given ring. If p = 0, then this identity assumes the form
x = x.
(6) xaa+"P = (xa )a + (xß)ß , where a , p E Der Q and a,ß EC.
Let Z)jm be the Lie subalgebra of Der Q consisting of all inner derivations
of Q . We choose a basis MQ for the right C-vector space 7>m and augment
it to a basis M of 7>m + D. We also fix a total order > in the set M such
that p0 > p for pQ E MQ and p E M\MQ. We then extend this order to the
set of all derivation words in M by assuming that a longer word is greater than
a shorter one and that words of the same length are ordered lexicographically.
By a regular word in M , we mean a word of the form A = ôx'ô2 ■ ■ ■ ôs™ such
that
(1) Ó:EM\M0 for /= 1, ... ,m,
(2) ôx < ô2 < ■ ■ < ôm , and
(3) s■ < p for i — 1, ... , m , if the characteristic of 7? is p > 0.
By means of the basic identities (l)-(6), any differential identity can be trans-
formed into the form <p(xi ' ) = 0 where
(1) <p(zr) is a generalized polynomial in distinct indeterminates z..,
and where
(2) A are regular words in M .
Kharchenko [4] has proved the following (actually for semiprime rings with
characteristic):
Theorem (Kharchenko). Let R be a prime ring. Suppose that cp(x¡') = 0 is a
differential identity for R, where A are distinct regular words. Then </>(z/y) = 0
is a generalized identity for R, where z, are distinct indeterminates.
Now suppose that R is a prime ring endowed with the involution *. Note
that * can be uniquely extended to Q in the following manner: Let a E Q
and let I E F be such that la ç R and al ç R. Define a* by the following:
a* : RI* —> RR via r*a* = (ar)* . It is obvious that a* E RF . We can verify
easily that for r E I, a*r* = (raf E R. So a* E Q as desired. We assume
henceforth that * is defined on the whole Q. We say that * is of the first kind
if a* = a for all a E C . Otherwise, we say that * is of the second kind.
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»DIFFERENTIAL IDENTITIES OF PRIME RINGS 255
By a ^-differential polynomial, we mean a generalized polynomial involving
noncommutative indeterminates which are acted by the involution * as well as
derivations of 7?, regarded as unary operations. Although, in general, * cannot
be extended to RF , we still allow the coefficients of a ^-differential polynomial
to lie in RF . The following is an example of a *-differential polynomial:
4>(x,y) = axx* '*a2y 2*a3 + bxyb2y 2x*b3,
where ax, a2, a3, bx, b2, b3 E RF and ôx, ô2 E 7>m + D.
For a *-differential polynomial cf>, we say that cf> — 0 is a ^-differential
identity for R if d> assumes the constant value 0 for any assignment of values
from R to its indeterminates.
The following basic identities are trivial for any rings with involution * :
(7) (xy)* = y*x*.
(8) (x + y)* = x* + y*.
For er e Der Q, we define o* as follows: For x E Q, x = ((x*)a)*.
It is easy to verify that a*, thus defined, is also a derivation of Q. The
following basic *-differential identity holds in any rings with involution * :
(9) (xaf = (x*){a'], where aeDerß.
An immediate generalization of (9) is the following:
(9)' (xSv"Sn)* = (jc'jWW-W), where sx,ô2,... ,8n E DerQ.
By means of (9)', we can push * inward or outward through derivations.
Bringing * outside of derivations, every »-differential identity can be trans-
formed to the form <f>(x¡ ', (xiA)*) = 0, where A are words of derivations
only, not containing any *. By using identities (l)-(8), we can further trans-
form each A into regular words as we did in Kharchenko's theorem. From
now on, unless specified otherwise, whenever we write cp(x¡ ', (xtJ)*), A are
always understood to be regular words.
Our main objective is to prove the following »-version of Kharchenko's the-
orem:
Main Theorem. Let R be a prime ring with involution *. Suppose that
(p(x¡■' ,(x¡ J)*) = 0 is a »-differential identity for R, where A are distinct reg-
ular words of derivations in M with respect to <. Then </>(z.. ,z*.) = 0 is a
^-generalized identity for R, where z.. are distinct indeterminates.
Now we prove an immediate generalization of Kharchenko's theorem, which
will be needed later. Suppose that for each indeterminate x-, we pick a basis
M{0'} for DiM and then augment MlQ'] to a basis M{,) for DiM + D. We also
fix a total order <(,) in the set M{l) such that p <(,> p0 for p0 e M^ and
p E M{i)\Mq} . We then extend this total order to the set of words in M(>)
and also define regular words in M(l) with respect to <(,), as we did before
for M with respect to < when we stated Kharchenko's theorem. If desired,
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256 CHEN-LIAN CHUANG
M(l), < can be made to coincide with each other. But in general, they are
completely arbitrary and independent of each other.
By means of the basic identities (l)-(6), any differential identity (withouta'"
involution *) can be brought to the form <p(xi ' ) - 0, where
(1) cp(ztj) is a generalized polynomial in distinct indeterminates z..,
and where
(2) for each i, A{l) are regular words in tV7(,) .
We have the following:
Generalized Kharchenko theorem. Let R be a prime ring. Suppose that
4>(Xj ' ) = 0 is a differential identity for R, where Av' are distinct regular words
in Mi with respect to <(i) for each i, j. Then cf>(zij) is a generalized identity
of R, where z are distinct indeterminates.
Proof. Let us assign arbitrarily certain fixed values from 7? to the indetermi-(1) A(l'
nates x2,x3, ... of cp other than xx . The resulting expression dy- '(x > ) = 0
is a differential identity involving the indeterminate xx only. Applying the orig-
inal Kharchenko's theorem to </5 = 0, we obtain that cfy (zXj) = 0 is a gener-
alized identity for R . Since the assignment of values from R to x2,x3, ... is
A<"completely arbitrary, <p(zXj ,x¿ ' )¡>2 = 0, the identity obtained by substituting
A(l» AUI
Zj. for xx' in 4>(x¡' ) = 0, is still a differential identity for 7?. ContinuingA(/)
this process, we can eventually replace all x¡ ' by x( . This completes the
proof.
If we let M{1), <(,) all coincide with M = M{X), <=<(l), then the above
generalized theorem specializes to the original one.
Back to the »-version, let M0, M, < be as explained in Kharchenko's
original theorem. Define tV/J = {p* : p E M0} and M* = {p* : p E M) . For
px , p2 E M, we define p* <* p2 if and only if px < p2. For a word of
derivations A = 5X62■ ■ ■ ôn , where ôt E M, we define A* = f5*f52 • ■ -S*. Note
that if A is a regular word in M with respect to < , then A* is a regular word
in M* with respect to <*.
Using the identity (9)', cp(xfi, (xt J)*) = 0, where A. are regular words in
M, can be transformed into cp(xf' ,(x*) ' ) = 0, where A. and A* are regular
words in M and M *, respectively.
II. Special »-differential identities
Let y/(xf' ,(x*)áj) = 0 be a »-differential identity for R, where A., A^
are simply words of arbitrary derivations, not necessarily regular words in M.
Then y/(xf' ,(x*)&J) = 0 is said to be special for 7? if y/(x¡ 'Ay A) = 0 is
a differential identity for 7?, where y¡ are new indeterminates distinct from
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»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 257
x¡. In other words, y/(xfJ ,(x*)A') = 0 is special for 7? if x¡ and x* can
be regarded as independent indeterminates. This generalizes the definition of
special generalized identities given in [7, p. 471].
Suppose that we use identities (l)-(6) to transform y/(xf', (x*)A') = 0 into
another identity d(xfJ ,(x*)A>) = 0, where A., A. are also words of arbi-
trary derivations (not necessarily regular in M ). Then the same procedure also
transforms y/(xfJ ,yf'J). = 0 into 6{xfJ ,yfJ)=0. Thus if y/(xfJ ,(x*f>) = 0
is special for R, then so is 6(xA ,(x*) ') = 0. We have thus shown that
the speciality of a ^-differential identity is independent of transformations using
identities (l)-(6).
If every »-differential identity of the form y/(xA', (x*)Aj) = 0, where A.,
A' are words of arbitrary derivations (not necessarily regular), is special for 7?,
then 7? is said to be special. The relationship between the speciality and our
Main Theorem is the following:
Lemma 1.7/7? is special, then the Main Theorem holds.
Proof. Suppose that <f>(xA ,(xA)*) = 0 is a »-differential identity, where A
are regular words in M. Using the identity (9)', <¡>(xA ,(xf')*) = 0 can be
transformed into cp(x¡ ', (x*) ' ) = 0, where A* are hence regular words in M*.
Suppose that R is special. Then <p(xi' ,yt■') - 0 is a differential identity for
7?, where y¡ are new indeterminates. By the generalized Kharchenko theorem,
(p(z¡j, w¡ .) = 0 is a generalized identity for R. In particular, <p(z¡ , z*.) — 0 is
a »-generalized identity for 7?.
The aim of this section is to prove the following generalization of Theorem
7 [7, p. 473]:
Proposition 1. Suppose that * is of the second kind and that C is infinite. Then
R is special.
The proof will be completed by a series of lemmas. First, we recall some more
definitions from [3]. Let Q' be the ring anti-isomorphic to Q with the same
additive group. That is, Q' is the opposite of Q. Let B denote the subring
of the tensor product Q <g>z Q (where Z is the ring of integers) generated by
the elements of the form 1 ® r, rigil, r e R. Elements of B are of the form
J2¡ r ® vi, where r., vt e R. For a E Q, ß = £, r¡ ®v¡ e Q ®z Ql , let
a • ß = ¿/ «W • *™ a E Q, let aL = {ß E B: a ■ ß = 0} . For V ç B, let
V = {a E RF : a • V = 0}. If A is an endomorphism of the abelian group
Q, we set ß = £\ r¡ ® vj. (This is well defined!) If f(x) is some linear
expression involving the variable x and if ß = J2, r¡ ®vtEB, then we set
f(x) • ß = Y,j Vjf(rjx) ■ Observe that if f(x) is identically zero on R, then so
is f{x) ■ ß .
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258 CHEN-LIAN CHUANG
We quote the following [3, Lemma 1]:
Lemma 2. Suppose that ai e RF (i = 1,2, ... , m). Then
777 / 77! \ -1-
Z"iC=(n«t) ■7=1 \l=l /
Suppose that cp(xj ', (xt ')*) = 0 is a »-differential identity, where A are reg-
ular words in M. cp(x¡ ', (xt')*) = 0 is said to be trivial if the »-generalized
polynomial <f>(zi., z*.) is trivial, or equivalently if the ordinary generalized poly-
nomial ^(z,.. ,j>..) (without *) is trivial.
A linear »-generalized identity f(x) is of the form
fix) = J2 aixbi + Ç cjx*dj = °.' j
where ai,bi,Cj ,d- E RF . f is trivial if and only if J2¡ a¡ ®c bi ~ ® and
X),- Cj ®c d= 0. A linear »-differential identity cfi(x) can be written in the
form
77, 777,
cp = d>(x\(xrr)=ET.a.k]xA,b,k)+EEc?^r;)*<).i k=l j /=1
where a¡ ,b\A ,c. ,d E RF and where A;, T are regular words in M.
Using the identity (9)', cp can also be written in the form
71, 777,
cp = 4>(x\(x*f>) = Y.T,4k)*V + ££c;V)r;<>,7 k=\ j 1=1
where V* are, certainly, regular words in M*. cp is said to be trivial if and
only if E"'=i a[,k) ®c bf] = ° for a11 ' and ^/=i ci" ®c df = ° for a11 J ■The following lemma is essentially the »-version of Lemma 2 [3].
Lemma 3. Suppose that every linear »-generalized identity of R is trivial. Then
so is every linear »-differential identity of R .
Proof. We need the following formula from [3, p. 158]: Let A = Sx ■ ■ âm be a
regular word in M such that Sx = S2 — ■■■ = Ss ^ Ss+X , where 0 < s < p if
ch R = p > 0. Let ß E B . Then
(1) (axAb) -ß = (a- ß)xAb + s(a ■ ßS')x'h"amb + ■■■ ,
where the dots above denote a sum of terms dx b in which A < S2 ■ ■ ■ ôm .
We define a partial order on ordered pairs of regular words in M . Assume
that A! , A,, r. , and T-, are regular words in M. We define (A,, I",) >
(A2, r,) if A, > A2 and Tx > T., or if A, > A., and Tx >T2.
Let M be a finite subset of M. Note that the set of all regular words in
M is well ordered under < . Hence the partial order < on word pairs defined
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»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 259
above is well-founded when restricted to ordered pairs of regular words in M .
Let J^(M) denote the set of all linear »-differential identities involving only
derivation words in M. J?(M) consists of the »-differential identities of the
form
££^X + ££^V^7 = oi k j I
where A., T are regular words in M.
Now suppose that
77, m j
m = £Efl/V*!fc)+££4V0H(/) = oi k=\ 7 l=\
is an arbitrary linear »-differential identity in S(M) ; that is, A(., T are
regular words in M. We may assume that (A,,Yx) is the leading word pair of
cp ; that is, A, > A. and Yx > Y for all i, j . We proceed by induction on the
leading word pair (A! ,YX) (under the partial order defined on word pairs) to
show that cp is trivial.
If (A,,r,) = (0,0), then <f> is simply an ordinary »-generalized identity
without derivations. By our assumption, cp must be trivial. So suppose that
(A1,r,)/(0,0). Let us first assume that A,/0. Obviously a\X), ... ,a\"']
can be assumed to be C-linearly independent.
We claim that it suffices to prove the case when nx = 1 : Suppose that nx > 1
is given. Let ß E f]"k'>2{a\ ) • Using formula (1), cp(x) • ß has a single term
(a\ ] ■ ß)x lb\ containing the regular word A, . If we have proved the case
when nx = 1, then cp(x) • ß is trivial and hence a\ ' • ß — 0. So a\ • ß — 0
for all ß E fl^^S ))± and' by Lemma 2, a[X) E J2l>->a[ 'C, a contradiction
to the C-linear independence of a\ , ... ,a" .
So we may assume nx — I . Suppose Ax = SXS2 ■■■ S , where Sx = S2 — ■• ■ =
ôs / ös+x and 0 < s < p if ch R = p > 0. Set A, = ô2 ■ ■ ■ Sm . Let px, ... ,pn
be all derivations in M other than <5, such that /j,A, , ... ,/i„A, occur in cp
as some A(. Let us say Ar = pxAx, ... ,Ar = pnAx . We also assume that
A, occurs in cp as Ak . In view of formula (1), the sum of terms of cp(x) • ß
containing x ' has the following form:
71 71 r,
,(fc) 0/7twA,A(7t) , X^fji) öwA,,(i)9k„
= 1 tV=1 i
s(a\X) • ßs')xA'b\X) + E£(<' • ß*)M? + £(<' • ß)^bl
If a\ • ß — 0, then the leading word pair of cp(x) • ß is strictly less than that
of cp. Since cp(.x) ■ ß E S(M), by our induction hypothesis, <p(x) ■ ß must be
trivial. So we have
(2) s(a\l) ■ ßd>)®c b\X) + J2K] ■ ß"') ®c K, + £(< ■ ß) ®c Kl = 0-i .k i
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260 CHEN-LIAN CHUANG
Since 0 < s < p in the case of ch 7? = p > 0, 5 is nonzero as an element of
C. Hence a[X) ■ ß ' can be linearly expressed in terms of a^ ■ ßM* and a{k ■ ß .
Using the linearity of ( ) • ßßx, () • ß , and introducing new notation, we have
a[X).ßs>+^2dT.ßß<+h.ß = 0.T
Since the right coefficients of (2) do not depend on ß, neither do dr and h .
This shows that the mapping
C:a[X).ß^a\X).ßai+J2dT-ßßT+h-ß,X
where ß ranges over B, is well defined. Choose I E F such that Ia\ ç R,
IdTCR (t= 1, ...,«), and Ih ÇR. Set T = B(l®I). If ß E T = B(I®I),
then both a[x) ■ ß and Ç(a[x) ■ ß) fall in 7?. Thus C is defined on the ideal
{a\ • ß : ß E T) of R and the range of this ideal under Ç is also contained in
R. Furthermore, for v E R,
C(v(a[X)-ß)) = C(a[X)-ß(l®v))
= a[X] ■ ßaA[l ® v) + J2dz ■ ß"1 (I ® v) + h ■ ß(l ® v)T
= vC(a{X,-ß).
Hence, by the definition of RF , there exists t E RF such that, for ß E T,
(a{xX).ß)t = aa\l)-ß) = a\X).ßSi+J2äz-ß'i' + h-ß.T
For x E R , we compute
(a[X) ■ ß)(xt) = ((a\X) ■ ß)x)t = (a\X) ■ ß(x ® l))t
= a(X) ■ (ß(x ® l)f +Y,dx- (ß(x ® O)"' + h ■ (ß(x ® 1))T
= (a\X).ßa>)x + (a[X).ß)xS'
+ £(^r • ßßlx + £(^r • ß)*"' + (h ■ ß)x,X X
and
(a[x) ■ ß)(tx) = ((a\X) ■ ß)t)x = (a\X) ■ ßä')x + £>r ■ ß>")x + (h ■ ß)x.X
Hence we obtain
(3) (a1/' • ß)[x, t] = (a[X) ■ ß)(xt - tx) = (a\X) ■ ß)xSi + ^(^r ■ ß)xM'.X
Now suppose that a[X) — ax,a2, ... is a basis for the subspace a] ■ C +
J2 d ■ C . Express d in terms of this basis:
dx = axax + ■ ■ ■ , ax E C.
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»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 261
Let ß E n/>2 afnT. Then dT-ß = aT(ax ■ ß). By (3), we have
(a[X) .ß)([x,t]-xS> -J2*xxß') =Q-
Hence [x, t] = x ' + J2r otxx^ . This contradicts with our choice of the basis
M. So we must assume that A, = 0 and Yx ̂ 0 .
Replacing x by x* in <j> and using the identity (9)' to bring * inside of
derivations, we have
71/ mi
I 7C=1 7 /=1
By a similar induction on regular words in M — {ô* : ô E M} with respect to
< , we can show that Y. — 0 . This is another contradiction. We have thus
shown that any linear »-differential identity of J^(M) is trivial.
Now let
*(*> -Et^A'bi}+££c,VT^ = o7 7C=1 7 1=1
be an arbitrary linear »-differential identity. Let M be the finite subset of M
consisting of all derivations occurring in A, and Y.. Then cp e J?(M). By the
result of the previous paragraph, cp must be trivial.
Lemma 4. Suppose that every linear »-generalized identity of R is trivial. If
cp(xA ,(xA)*) = 0, where A are distinct regular words in M, is a multilinear
^-differential identity for R, then cp(zj.,yij) = 0, where z.., y¡¡ are distinct
indeterminates, is a multilinear generalized identity of R.
Proof. If cp involves only one indeterminate, say xx , then cp = cp(xx ', (xx ')*) is
trivial by Lemma 3. Hence cp(zXj ,yXj) = 0 is also a trivial generalized identity
for 7?.
Now suppose that cp involves more than one indeterminates, say xx, ... ,x .
Let us assign certain fixed values from R to x2, ... ,x . The resulting identity
thus involves the indeterminate xx only and hence must be trivial by Lemma
3. As in the previous paragraph, we can replace xxJ , (xx')* by new indeter-
minates Zj ■, yXj to obtain a generalized identity. Since the values assigned to
x2, ... ,xn are completely arbitrary, cp(zXj ,yXj ,xf' ,(xf')*)(>2 = 0, the iden-
tity obtained by replacing xx' , yx' in cp = 0 by z. , yx , respectively, also
holds on R. Repeating the same argument for x2, ... ,xn, the desired result
follows.
By assigning fixed values to all indeterminates but one, we can reduce a given
identity to an identity involving only one indeterminate. This technique, used
in proving the generalized Kharchenko theorem and also in the proof above,
will be used frequently throughout this paper.
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262 CHEN-LIAN CHUANG
Lemma 5. Suppose that is of the second kind. Then every multilinear
^-differential identity of R is special.
Proof. By Theorem 7 [8, p. 473], every multilinear »-generalized identity of R
is special. In particular, every linear »-generalized identity is special and hence
must be trivial by Lemma 1.3.2 [1, p. 22]. The result now follows from Lemma
4.
Lemma 6. Suppose that * is of the second kind and that ch 7? = 0. Then R is
special.
Proof. Let cp be a given »-differential identity. Replacing each indeterminate
x( by 2x;, 3x;, ... and using the Vandermonde determinant, we can solve
the homogeneous components of cp. Hence each homogeneous component of
cp is also a »-differential identity of R. It suffices to show that each homo-
geneous component of cp is special. Replacing cp by one of its homogeneous
components, we may assume from the start that cp is homogeneous in each
indeterminate.
Let 6 be the multilinearization of cp. By Lemma 5, 6 is special. Identifying
the indeterminates in 6 properly, we obtain ncp, where « is a nonzero integer.
Hence ncp is also special and so is cp, since chT? = 0.
Lemma 7. Suppose that * is of the second kind and that ch R = p > 0. If C is
infinite, then R is special.
Proof. By the technique used in proving Lemma 4 (and also the remark fol-
lowing), it suffices to prove the case when cp involves only one indeterminate,
say x. We proceed by induction on the degree of cp. If deg <p = I, then cp is
linear. The result follows from Lemma 5. So we assume that degcp > 1 . As the
induction hypothesis, we also assume that the result holds for any »-differential
identity with less degree. Bringing * inside of derivations by identity (9)' and
suppressing words of derivations for simplicity of notation, we write cp(x ,x*)
for cp(x ', (x*) '). Now consider
0(x,x*,j;,/) = cp(x + y,x* +y*) - cp(x,x*) - cp(y ,y*).
By the induction hypothesis (and the remark following Lemma 4), 6 is special
for R . Now regard x , x*, y , and y* in 6 as independent indeterminates as
granted. Set x = x, x* = 0, y = 0, and y* = x* in 6 :
6(x,0,0,x*) = (p(x,x*) - <p(x,0) - (p(0,x*).
Thus cp(x,0) + cp(0,x*) is also a »-differential identity for R. We also have
the identity 0(x,x*) = 6(x,0,0,x*) + cp(x,0) + cp(0,x*). Since 6(x,0,0,x*)
is special for 7?, it suffices to prove that cp(x,0) + 0(0, x*) is special. This is
equivalent to showing that both <p(x,0) = 0 and 0(0,x*) = 0 are differential
identities for 7?.
Set Ç(x) = 0(x,O) and r¡(x*) = 0(0,x*). For positive integers t > 0, let
C,(x) and t].(x*) denote the sums of those monomials with degree t in Ç(x)
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»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 263
and in t](x*), respectively. Pick sufficiently but finitely many a e C such that
a* = a . Let 7 be a nonzero »-ideal of 7? such that apI ç R for those finitely
many a E C we have picked. Then Ç(apx) + rj(apx*) vanishes on 7. Since
ap are constants of any derivations, we can use the Vandermonde determinant
to solve the homogeneous components of C(c/x) + t\(apx*) = 0. So we have
that, for each t > 0, Ç,(x) + t]t(x*) vanishes on 7.
We claim that there exists a E C such that apt ^ (a*)pl : Pick ß e C such
that ß* ± ß . For s E C such that s* = s, set a(s) = s + ß . Observe that
a(s)* =s + ß* =u ß'-ßa(s) s + ß s + ß
Hence if s ^ s', then a(s)*/a(s) ^ a(s')*/a(s'). Since the equation ypt = 1
has only finitely many solutions in the infinite field C, there exists a symmetric
element seC such that a(sf ¿ (a(s)*)pl.
Pick a E C such that apl ^ (a*)pl as claimed above. Let 7 be a nonzero
»-ideal of R such that aJ ç 7?. Then Çt(apx) + nt((apx)*) = 0 holds for
xeIJ . Hence for x € 77 , 0 = ap'(Ct(x) + t]t(x*)) - (Çt(a"x) + r¡t((a"x)*)) =
(apl - (a*)pl)r]t(x*). Thus r¡t(x') = 0 holds on 77. Similarly, £;(x) = 0
also holds on 77. Note that Çt(x) = 0 and r\t(x) = 0 are simply ordinary
differential identities without the involution *. By the remark on p. 74 in [4],
Ct(x) = 0 and ^(x) = 0 also hold on R. This completes the proof.
Proposition 1 now follows from Lemma 6 and Lemma 7.
One might wonder whether or not the assumption that C is infinite can be
eliminated in Proposition 1. The answer is negative, as the following example
shows.
Example. Fix a positive integer n > 1 . Let <P be a finite field of characteristic
p > 0 with p " elements. For a E O, define a — ap . - is an involution of
the second kind on O. The symmetric elements of <I> satisfy the polynomial
identity s = s. But the »-identity (x + x) = x -l- x is obviously not special.
For a noncommutative example, we can take any k x k matrix ring <i>k
(k > 1) with transpose involution * defined as follows: For a¡¡ GO (i,j =
1, ... ,k), (a¡j)* - (ccjj). For an infinite-dimensional example, let V be an
infinite-dimensional vector space over O. Pick a Hermitian bilinear form ( , )
on 0F. That is, ( , ) satisfies the condition iav ,ßw) = aß(v ,w) for a,ßE
<t> and v ,w E V . Our desired ring is the set consisting of all continuous finite
rank linear maps in End^F). (See §111 for definitions.)
Actually, we can prove that any counterexample to Proposition 1 must be of
the form described above.
III. »-RINGS HAVING MINIMAL ONE-SIDED IDEALS
We digress here to generalize some basic theorems on rings having minimal
one-sided ideals to their »-versions.
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264 CHEN-LIAN CHUANG
First, we recall some basic definitions. Let D be a division ring. Suppose
that D V is a left vector space over D and IVD , a right vector space over D.
If there exists a nondegenerate bilinear form ( , ) : V x IV ^ D, then D V and
IVD are said to be dual with respect to ( , ).
Let A(V) = Hom(DV,DV) and A(W) = Hom(WD,IVD). For a E A(V)
and b E A(W), if (va,w) — (v ,bw) holds for all v E V and for all w E W,
then we call b an adjoint of a . Not every element a E A(V) has an adjoint.
But if a G A(V) does have an adjoint, then that adjoint must be unique and
will be denoted by a*. a E A(V) is said to be continuous if a has an adjoint
a* E A(1V). The set of all continuous elements a E A(V) forms a ring and
is denoted by LW(V). Let SW(V) = {a E LW(V): dimD(Ka) < 00}. For
u,, ... ,vn E V and wl,... ,w E IV, the linear transformation a E A(V)
defined by va = (v ,wx)vx + ■ ■ ■ + (v ,wn)vn is in SW(V) and has the adjoint
a* E S y (IV) given by a*w — wx(v x ,w ) + ■■• + wn(vn ,w). Conversely, any
a E SW(V) can be represented in the form described above. The following two
basic facts can be found in [2].
Fact 1. Let vx, ... ,vn E V be 7>independent. Then there exist D-indepen-
dent vectors ux, ... ,unE IV such that (il , m .) = ôt.for /, j — 1,2, ... , n .
Fact 2. Let VQ and IVQ be finite-dimensional subspaces of D V and H^ , respec-
tively. Then there exist finite-dimensional subspaces Vx 2 VQ and Wx D IV0 of
D V and IVD respectively such that the bilinear form ( , ) restricted to Vx x Wx
is nondegenerate. (Hence dim^ = dimH^ by Fact 1.)
For a prime ring R, let Soc(7<) be the sum of all minimal right ideals of
7? . It is well known that Soc(7?) is also the sum of all minimal left ideals of R
and that Soc(7?), if nonzero, is the unique minimal ideal of 7? . We have the
following representation theorem for prime rings with nonzero socle.
Jacobson theorem. Let D V and IVD be dual spaces over a division ring D with
respect to the nondegenerate bilinear form ( , ). Let R be a subring of A(V)
such that SW(V) ç R ç LW(V). Then R is a primitive ring with Soc(7?) =
SW(V).
Conversely, given a prime ring R with Soc(7?) / 0, we can find a division
ring D and a pair of dual spaces DV and IVD such that SW(V) ç R c LW(V).
The following fact can be found in [4, p. 171] or Proposition 7, [6, p. 98].
However, we give here a new proof using the density theorem, which is more
related to our next proposition.
Fact 3. Suppose that SW(V) CRC LW(V). Then RF = A(V).
Proof. Let h e RF . Since SW(V) is the unique minimal ideal of R,
Sw(V)h ç R. We define h E A(V) as follows: For v¡ E V and ri E SwiV),
h: Y,viri ^ 12vi(rih). Note tnat r,b G R and hence vf.rfi) E V. We
must show that h is well defined: Suppose J2vjr¡ - 0- Since r g SW(V),
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»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 265
dimD(X); Vr/) < oo. Choose r e Sw(V) such that r restricted to X,, Vr¡ is
the identity map. Then r¡r = r¡. Hence
5>/W = !>2vttrir)h) = T.viWrh)) = £¥()('*) = °-
Via this natural embedding /i —► h , we have 7<F ç /1(F).
Conversely, suppose that h E A(V). In order to prove that h E RF, we
must show that Sw(V)h ç R : Every element a G S^K) can be represented
by the form va = ^2¡(v ,w¡)v¡., where v ,v¡ E V, w¡ G IV. Hence vah =
J2]iv ,u;(.)(t;(./z). Then a/2 g SwiV) E R as desired.
The following proposition is interesting in itself.
Proposition 2. Suppose that SwiV) CRC LW(V). Then Q = LwiV), where
Q is the two-sided quotient ring of R.
Proof. Suppose that h E LwiV). Then hSw(V) C SW(V) ç R and Sw(V)h çSW(V)CR. So hEQ.
Conversely, since Q ç R , by Fact 3 above, we may assume QcA(V). Let
hEQ. In order to show that h E LW(V), we must find its adjoint h* E A(W).
Let SW(V)* = {r*:rESw(V)}, R* = {r* : r E R} , and LW(V)* = {r* : r E
LW(V)}. Observe that SW(V)* = SV(IV) and LW(V)* = LV(W). Hence
SV(W) ER* E LV(W). Let F* be the filter of nonzero two-sided ideals of R*
and let F.R* be the right quotient ring of R* relative to the filter F*. By a
symmetrical version of Fact 3 for right quotient rings, we have F.R* — A(W).
Given h E Q, we define h* : SV(1V) —► R* as follows: For t E SW(V), set
h*(t*) = (ht)*. Since h E Q, we have hSw(V) ç R as well as Sw(V)h ç R.
So ht E R and hence (ht)* exists. Thus h* is well defined.
For r E R and t E SW(V), h*(t*r*) = h*((tr)*) = (h(tr))* = ((ht)r)* =
(ht)*r* = h*(t*)r*. So h* is a right 7?*-homomorphism and hence h* EF, R* =
A(W). To prove that h* is the adjoint of h , we must verify that (v , h*w) =
(vh, w) for any v e V , w e IV.
Using the fact that SV(IV) = SW(V)* and the Jacobson density theorem,
there exists / G SW(V) such that t*w = w . Then we compute
(v ,h*w) = (v , h*(t*w)) = (v ,(h*t*)w) = (v ,(ht)*w)
= (v(ht),w) = ((vh)t,w) = (vh,t*w) = (vh,w).
So h* is really the adjoint of h and hence h E LW(V) as desired.
The following generalization of Theorem 7.3.16 [9, p. 269] is the »-version
of the Jacobson density theorem.
Proposition 3. Suppose that D V and IVD are dual spaces with respect to the
nondegenerate bilinear form ( ,). Let v,,..., vs, v'x,...,v's e V, and
ux, ... ,ut, u'x, ... ,u't E IV be such that {vx , ... ,i\} is D-independent in
V and {m, ,... ,u,} is D-independent in IV. Then there exists a E SW(V)
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266 CHEN-LIAN CHUANG
such that v¡a = v\ (i = I, ... ,s) and a*u — u'¡ (j = I, ... ,t) if and only if
(v'i. Uj) = (vj,u'J) for i = I, ... ,s and j = I, ... ,t.
Proof. The necessity is easy: Suppose that there exists a G SW(V) such that
vta = v'j (i = I, ... ,s) and a*u. - u. (j = I, ... ,t). Then
(v'i, Uj) = (v(a, Uj) = (v¡,a*Uj) = (v¡, u'ß.
To prove the sufficiency, by Fact 2, there exist finite-dimensional subspaces
VQ, IVQ of V, IV respectively such that vl,...,vs, v'x,...,v's e Vq and
ux, ... ,ut, u'x, ... ,u't E IVQ and such that the bilinear form ( , ) restricted
to VQ x IVQ is nondegenerate. Also VQ and IVQ are of the same dimension, say
of dimension n . Augment {vx, ... ,vs} to a basis {vx, ... ,vs, vs+x, ... ,vn)
for V0.
By Fact 1, there exist x,, ... ,xt E VQ such that (x¡,u¡) = <5( for i,j =
1, ... ,t. For each s < k < n , define v'k = Y^'/=i(vk > u'/)xi ■ Then for s < k < n
and for j — I, ... ,t,
K'w,)= ( ¿K »"/)■«/»MyJ =iVk,u'j).
Together with the originally given v[, ... , v's, we have that
(v., u ) = (v¡ ,u'j) for i - I, ... ,n and for j - I, ... ,t.
By Fact 1 again, pick a basis {wx, ... ,wn} for IVQ such that (<7.,w.) =
S¡j for i,j = 1, ... ,n. Now we define our desired a G S[V(V) by va =
2~2"=\(v >wj)v'j ■ Observe that v¡a = (v^w^v^ = v\ (i = I, ... ,n). So it
suffices to prove that a* Uj = «' for j = 1,...,/.
Given Uj (1 < j < t), compute (w(. ,a*w.) = (v¡a,Uj) = (v'^Uj) for / =
1, ... ,n . Since v(' (/ = 1, ... ,n) are chosen so that (v't,Uj) = (v¡,u'j) for
j = 1,...,/, we have (il ,a*^) = (w(', uß = (v¡, uß . So (v¡,a*u} - u'ß = 0
for /' = I, ... ,n. Hence (VQ,a*u. - u'ß = 0. Note that a* is defined by
a*u = ¿Z,"=i wi(v'i>u) € ^o ^or any ueW . So, in particular, a*«. G W^ and
hence also a* m - «'. G IV0 . Since the bilinear form ( , ) restricted to VQ x IVQ
is nondegenerate, we have a* u. - u. = 0 as desired.
Let D be a division ring endowed with involution * and let DV be a left
vector space over D. V can also be regarded as a right vector space by the right
scalar multiplication defined by va = a*v , where v E V and q G D. Note
that A(DV) = A(VD). For this reason, the action of a G A(VD) on VD will also
be written on the right-hand side. Suppose that there exists a nondegenerate
bilinear form on D V x VD . Then D V is said to be a self-dual space with respect
to ( , ).
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»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 267
The »-version of the Jacobson theorem is the following well-known:
Kaplansky theorem [ 1, p. 17]. Let R be a prime ring with involution * and with
Soc(Tv) t¿ 0. Then there exists a vector space V over a division ring D endowed
with involution, which is self-dual with respect to a Hermitian or alternate bilinear
form ( , ) in such a way that
1. SV(V)CRCLV(V),
2. the * of R is the adjoint of R relative to this bilinear form ( ,).
Furthermore, the Hermitian and alternate cases are mutually exclusive, occur-
ring according as there is, or is not, a minimal symmetric idempotent e = e =*
e .
Conversely, any R such that SV(V) ç R ç LV(V) and such that R* = R
must be a prime ring with involution * and with Soc(Tv) = SV(V) ^ 0.
Proposition 4. Suppose that R is a prime ring with involution * and that f
is a ^-generalized polynomial. If f vanishes on a nonzero ideal of R, then f
vanishes on Q, the two-sided quotient ring of R.
Proof. Suppose that / vanishes on a nonzero ideal I of R. Replacing 7 by
7 n 7*, we may assume from the start that 7 is a »-ideal of R. We may also
assume that / is nontrivial. By Theorem 9 [8, p. 477], 7 satisfies an ordinary
nontrivial generalized polynomial identity (without * ) and hence so does 7?.
By Martindale's theorem, RC is a primitive ring with nonzero socle. By Ka-
plansky's theorem, SV(V) ç RC ç Ly(V) for a self-dual vector space V over
a division ring D endowed with involution. By Proposition 2, the two-sided
Martindale quotient ring of RC is LV(V). Note that the two-sided Martin-
dale quotient ring LV(V) of RC includes the two-sided Martindale quotient
ring Q of 7? as a subring in a natural way. It suffices to show that / van-
ishes on LV(V). Note that Soc(TvC) = SV(V). By Proposition 3, it suffices to
show that / vanishes on Soc(7?C): Suppose that / vanishes on Soc(7?C).
Let qx,q2, ... e LV(V) and let v E V. In computing vf(qx ,q2, ...) = v ,
we need only finitely many relations of the form uq. = u , q*w — w', where
u,u ,w ,w' e V. Obviously, (u ,w) = (uqt,w) - (u,q*w) = (u,w'). Propo-
sition 3 asserts that there exists qj E Soc(RC) such that uq¡ - u and (q¡)*w =
w . Hence v = vf(qx ,q2, ...) = vf(qx , q2 ,...) = 0 .
If C is finite, then there is a nonzero »-ideal 7 of R such that aJ ç R
for all a E C. Then 7 3 77C. But 77C is a nonzero ideal of RC and
hence must contain the minimal ideal Soc(TîC). So / vanishes on Soc(7?C)
as desired.
Now let C be infinite. If * is of the second kind, then R is special by
Proposition 1. The assertion follows from the well-known corresponding result
for ordinary generalized polynomial identities (without *). So we may assume
that * is of the first kind. We proceed by induction on the height of /. Pick
sufficiently but finitely many a E C and let 7 be a nonzero »-ideal of R such
that aJ C I for these a picked. Replace each indeterminate x in / by ax
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268 CHEN-LIAN CHUANG
for these a picked. Then the resulting »-generalized polynomials vanish on 7 .
Note that these a E C picked are symmetric and hence can be moved out of
operation. By the Vandermonde determinant argument, we can solve these
»-generalized identities for the homogeneous parts of /. So each homogeneous
part of / vanishes on 7. It suffices to show that each homogeneous part of
/ vanishes on Q. Note that the height of each homogeneous part of / is not
larger than that of /. Replacing / by one of its homogeneous parts, we may
assume from the start that / is homogeneous in each indeterminate it involves.
Assume that the height of / is 0. Then since / is homogeneous, / must
be multilinear. It is obvious that / vanishes on 7C. Since 7C, a nonzero
ideal of RC, must contain the minimal ideal Soc(7?C), / also vanishes on
Soc(TvC) as desired. So we assume that the height of / is larger than 0. As
the induction hypothesis, we also assume that the assertion of the proposition
holds for any »-generalized polynomials with less height.
Given any indeterminate x in /, we suppress all indeterminates other
than x and write f(x) for / for simplicity of notation. Now consider
g(x + y) = f(x + y) - f(x) - f(y), where y is a new indeterminate. Since g
is of less height than f, g must vanish on Q by our induction hypothesis. So
f(x + y) — f(x) + f(y). We have thus shown that / is additive on Q with
respect to each indeterminate it involves.
Now write / = f(xx, ... ,xn), where xx, ... ,xn are all the indeterminates
/ involves. Set x( = ¿Zjrfotf, where rj0 G 7 and af E C. Using the
additivity of / on Q, we compute
a*,,*,.^»/Íew.-.ew)
7l .In
= £ (^)'"---(c^)h"f(r{X),...,^),
71 .jn
where /?, = the x,-degreeof / (/'= 1,...,«). Since rf e I, f{r^, ... ,rfj)
= 0 and hence f(xx , ... ,xn) = 0. But x( = J~j.r^oA1' are typical elements of
IC. So f vanishes on 7C. Since 7C, a nonzero ideal of RC, must contain
the minimal ideal Soc(7?C), / vanishes on Soc(7?C) as desired.
One might be wondering whether the two-sided Martindale quotient rings
respectively of R and of 7?C are equal or not. If R satisfies a nontrivial
polynomial identity (over C), then 7?^, Q, the left Martindale quotient ring
of RC, and the two-sided Martindale quotient ring of RC are all equal to
RC . But, as the following example shows, this is actually false in general, even
if 7? satisfies a nontrivial generalized identity.
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»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 269
Example. Let D be a (commutative) principal ideal domain and let C be its
quotient field. Suppose that CV is an infinite-dimensional vector space over
C . Fix a basis ¿& = {vx,v2, ...} of CV. For v - J2¡ aiv¡ » u — J2¡ ßtvi G v >
where vi E 3§ and where a¡, ßt E C vanish for all but finitely many i, we
define (v ,u) = J2¡a¡fi¡. Then ( , ) is a nondegenerate bilinear form on VxV.
As usual, let A(V) = End(rF). For a G A(V) such that v.a = £.■<*,,«,,t- l J 'J J
where i>(, VjESS, ai. G C, the adjoint of a, if it exists, is defined by
w a* = IZ/a/,ï,(- Hence, LV(V) consists of all a G /1(F) such that for each
u ■ G «^ , (v(a, Vj) = 0 for all but finitely many i, and SV(V) consists of all
a E A(V) such that v¡a - 0 for all but finitely many i. Let D^ consist of all
aEA(V) suchthat (via,vj)ED for all v^VjESS . Define R = Sv(V)nDoo.
Then RC = SV(V). Also, ideals of R assume the form dR for some d E D.
Hence the left Martindale quotient ring RF of R and the two-sided Martindale
quotient ring Q of R are equal to {a/d: a E D^ and 0 / d E D) and
RF r\Lv(V), respectively. So, if D is chosen so that D ^ C, then A(V) ^ RF
and LK(F)^ß-
IV. Proof of Main Theorem
We begin with the following.
Lemma 8. Suppose that R satisfies a nontrivial »-differential identity. Then R
satisfies a nontrivial generalized identity (without *).
Proof. By linearization, we may assume that R satisfies a nontrivial multilin-
ear »-differential identity 0(x|' ,(xAf) = 0. Assume toward a contradiction
that R does not satisfy any nontrivial generalized identities (without *). By
Theorem 9 [8, p. 77], R does not satisfy any nontrivial »-generalized identities
(without derivations) either. In particular, any linear »-generalized identities
of 7? are trivial. Hence Lemma 4 applies and gives that cp(z¡.,yj.) — 0 is a
multilinear generalized identity of R . Since 0(xf;, (x;A;)*) = 0 is not trivial,
0(z. ,y..) = 0 cannot be trivial either. This is absurd.
If every »-differential identity of R is trivial, then our Main Theorem holds
trivially and there is nothing to prove. So from now on, we assume that 7?
satisfies a nontrivial »-differential identity. By Lemma 8 above, 7? satisfies a
nontrivial generalized identity.
We recall the following [4, p. 68]:
Fact 4. Assume that R satisfies a nontrivial generalized identity. Let p E
D + Dm% be such that p(C) = 0. Then p E 7>nt.
If C is finite, then p(C) = 0 for any p E D + DmX. Hence by the identity
(3), any »-differential identity can be brought to a »-generalized identity without
derivations. So our Main Theorem holds trivially in this case. Thus we may
assume henceforth that C is infinite. By Proposition 1 and Lemma 1, we can
further assume that * is of the first kind.
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270 CHEN-LIAN CHUANG
We recall the following definition [4, p. 64, fourth line from the bottom]: For
px, ... ,pnE D + Dinl, we say that px, ... ,pn are strongly independent if, for
any a; G C, J2p¡cxi E 7>m implies a, — a, = • • • = an = 0.
Lemma 9. Suppose that px, ... ,pn E D + DmX are strongly independent and
that ax, ... ,an E RF . If for all a E C, apl a, +-h afinan = 0, then ax =
••• = a„ = 0. '
Proof. Suppose not. Without loss of generality, we may assume that a, ^ 0.
Choose a basis vx= ax, v2, ... for the subspace ¿~2"=x atC. Express a( (i =
I, ... ,n) in terms of the basis vx,v2, ... : a, = vx and aj = ßtvx + ■ ■ ■ for
/ > 2 . Substituting these expressions into ap'ax-\-Yap"an = 0 and collecting
terms according to vx ,v2, ... , we obtain
(a"' + aßlß2 + aßiß3 + ■ ■ -)vx + (-)v2 + ■ ■ ■ = 0.
Since vx ,v2, ... are C-independent, aMl + aßlß1 + aPlß3 + ■ ■ ■ = 0 follows.
Hence the derivation px + p-,ß2 + p3ß3 + ■ ■ ■ vanishes on C and, by Fact 4,
must be in 7>nt. This contradicts the strong independence of px,p2, ... ,pn .
Lemma 10. If cp is a linear »-differential identity, then the Main Theorem holds.
Proof, cp may be written in the form
77,
, ,, A, . A,,*, V^V^7 (k) A, .(A:) , (k), A,,»,(A'h0 = 0(x ,(x ) ) = 2^£(ß,■ x b,- +c, (x ) d\ )'
I k=\
where A; are distinct regular words in M. Assume that A, is the greatest
among A(.. We prove by induction on A, that if cp(x ', (x ')*) vanishes on a
nonzero ideal of R, then cp(y¡,y*) = 0 is a »-generalized identity on R , where
y are new distinct indeterminates.
If A, =0, then 0 is simply a »-generalized polynomial (without deriva-
tions). The result follows from Proposition 4. So let us assume that Ax ^ <Z.
As the induction hypothesis, we also assume that the assertion of this lemma
holds for any linear »-differential identity whose leading word is strictly less
than A, .
Write A, = àfô(2i] ■ ■ ■ and assume that â\i] = o{2] = ■■■ = ô{s'] ¿ ô^ ,
0 < si < p (p = ch R). We also set Ä, = Ô{2]S{3] ■■■ . Let ß EC. Using the
formula ( 1 ) (in the proof of Lemma 3) and the fact that a* = a for any a E C,
we have
af)(ßx)A'bf) = af'ßx^b^+s^ß^x^bf' + ■■■
and
cf \(ßxf')*d\k) = cf]ß(xArdt]+s,c< V'^Vtff' + ■ ■ ■ -License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 271
Hence
<Kßx) = ß £ 5>í* >A(* >+c\k\xAr df])i k=\
V^V^ n&\ t (le) A,,(fc) (k), Aíx* ,(7C),+ ££57^ ' («, x 'b\ ' + c) \x ') d]') + ■■■
i k = \
oj.1 \ , V^V^ os\'] i (k) ^iu(k) , (*■')/ Ä,.» Ak). ,= ß<t>(x) + 2^l^siß (ai x b, +c)'ix ) d) ') + ■■■•
7 fe=l
Let 7 be a nonzero ideal of 7v on which 0 vanishes. Pick a nonzero ideal 7
of 7? such that 7 ç 7 and ßJ Q I. Then epißx) - ßcpix) vanishes on 7.
Note that the leading word A, of 0(/3x) - /30(x) is strictly smaller than A, .
So the induction hypothesis applies.
We want to collect the terms in epißx) - ßcpix) which contain the regular
derivation word A, : Let px, ... ,pn be all derivations in M other than S\
such that pxAx, ... ,pnAx occur in 0. Let us say pxA, = Ar , ... ,pnAx - Ar .
Then the sum of those terms in 0(/3x) - /30(x) containing A, is of the form
sy"f2(a[k)xAlb\k) + c\k)ixArd[k))k=\
+£rf£(<)^,^)+<',(^r<,)VT=l \k=\ J
By the induction hypothesis,
s^^yb^ + c^y*^)k=\
+ £^Í£(<)<", + <V<))Uor=l \A: = 1 /
is a »-generalized identity for 7?. Since this holds for any ß E C and since
ô\], px, ... ,pn are strongly independent, by Lemma 9,
77|
El (*■') L.(k) , (k) * ,(7Ck „(a, ybx +c\ 'y d\') = 0
k=\
is a »-generalized identity for R . Hence
71,
i V^Y^/ (*) A,,(A') , (Ar), A,,* ,(/,').^ = ££K- x bi +c) (* ) di )7>2 7C=1
also vanishes on 7 . The leading word of 0, is obviously smaller than that of
0. So by the induction hypothesis again,
££(^Vf) + cfV;^)) = oi>2 k = \
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272 CHEN-LIAN CHUANG
is also a »-generalized identity for R. Combining this with the identity
¿(ÍVÍ' + ÍVÍVo,7C=1
the desired result follows.
Using a proof similar to that of Lemma 4, we have
Lemma 11. 7/0 isa multilinear »-differential identity, then the Main Theorem
holds.
Using Lemma 11 and arguing as in the proof of Lemma 6, we have the
following
Lemma 12. If chR — 0, then the Main Theorem holds.
By Lemma 11, we may further assume henceforth that ch R = p > 0, where
p is a fixed prime.
Since 7? satisfies a nontrivial generalized identity, by Martindale's theorem
[7], the socle of RC is nonzero. Set a = Soc(7<C). By the Kaplansky theo-
rem, there exists a self-dual space DV over a division ring D endowed with
involution such that
SviV) E RC ç LviV)
and such that o — SviV). Note that VRC is an irreducible right 7?C-module
and D = Hom(l/RC, VRC). Since Soc(7?C) ^ 0, all the irreducible right 7?C-
modules are isomorphic. The division ring D is also unique up to isomorphism
and is thus called the skew field of RC . Note that, by [7], D is finite dimen-
sional over C.
We need the following from [4, p. 74]:
Fact 5. Let 7 be a nonzero ideal of R . Then ICP 2 o .
Note that the coefficients of »-differential polynomials of R are allowed to
lie in RF . Since the left Martindale quotient ring of RC includes RF as a
subring in a natural way and since, by Proposition 2, both the left Martindale
quotient ring of RC and the left Martindale quotient ring of Soc(TvC) are
equal to A(V), any »-differential polynomial of R can be naturally regarded
as a »-differential polynomial of Soc(7?C). The following lemma is stated in
this sense.
Lemma 13. 7/0 isa ^-differential identity of R, then cp is also a ^-differential
identity of a.
Proof. We prove by induction on the height of 0 that if cp vanishes on a
nonzero *-ideal I of R, then cp vanishes on JCP for some nonzero ideal J of
R . Our desired result will follow immediately from Fact 5.
As in the proof of Proposition 4, pick sufficiently but finitely many a E C .
Let 7 be a nonzero »-ideal of R such that aJ El for those a picked. Replace
each indeterminate x in 0 by apx for those a E C picked. Note that the
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»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 273
resulting »-differential polynomials after such replacement vanish on 7 . Note
that ap are constants of derivations and hence can be moved out of derivations.
Using the Vandermonde determinant argument, we can solve these »-differential
identities of 7 for the homogeneous parts of 0. So each homogeneous part
of 0 vanishes on 7 . Note that the height of each homogeneous part of 0 is
less than or equal to that of 0. Replacing 0 by one of its homogeneous parts
(and 7 by 7 ), we may assume from the start that 0 is homogeneous in each
indeterminate it involves.
If the height of 0 is 0, then 0 must be multilinear. If 0 vanishes on 7, then
obviously 0 also vanishes on ICP as desired. So we assume that the height
of 0 is larger than 0. As our induction hypothesis, we also assume that the
assertion holds for any »-differential polynomial of less height.
Let x be an indeterminate occurring in 0. Suppressing indeterminates other
than x, write 0 = 0(x) for simplicity of notation. Let y be a new indetermi-
nate not occurring in 0. The »-differential polynomial 0(x +y) - 0(x) - cpiy)
also vanishes on 7. Since the height of 0(x + y) - 0(x) - 0(y) is strictly less
than that of 0, by our induction hypothesis, there is a nonzero ideal 7 of R
such that 0(x + y) - 0(x) - cpiy) vanishes on JCP . We have thus shown that
0 is additive on JCP with respect to x .
Let xx, ... ,xn be all indeterminates occurring in 0. Write
0 = 0(X,, ... ,xn).
By the paragraph above, for each i (/' = 1, ... ,n), there exists a nonzero ideal
7( such that 0 is additive on JtCp with respect to x;. Set K = 7 n (f]"=i J,) ■
Assume that the x(-degree of 0 is h¡. Set x( = X^, rj aJ where r. G K and
a- E Cp . Using the additivity and the homogeneity of 0, we compute
*E']'X"EW.Y.AA
=h'iZ(A!)AA:f--A:,ArA':,rj:,...,r_)n >
J.)n
= 0.
Hence cp vanishes on KCP as desired.
By the Jacobson-Noether theorem, the skew field D of R possesses a maxi-
mal subfield O which is separable over C . <X> is hence a primitive extension
of C. Assume that <P = Civ), where v is a primitive element of O over C
satisfying the minimal polynomial
k/(*) = £<***
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274 CHEN-LIAN CHUANG
By the separability of O over C , f (v) = Y.k kakvk~x / 0 .
Any derivation p on C can be uniquely extended to <P (=C(v)) by defin-
ing
Now consider the ring cr ®c O. Any derivation /j on o can be extended to
o ®c O by defining
(r <g> a)" = r" 0 a + r ® a'',
where r G a and a G O. The involution * on a can also be extended to
cr ®c <P by defining
(r ®a)* = r* ®a,
where r E a and a E O.
Since the left Martindale quotient rings of a is a subring of the left Martin-
dale quotient ring of o®c<t> in a natural way, by means of the above extensions
of derivations and involution on o to a®c<&, any »-differential polynomial of
a can be naturally interpreted as a »-differential polynomial of a ®c <X>. The
following lemma is stated in this sense.
Lemma 14. If cp is a ^-differential identity on o, then cp is also a »-differential
identity on o ®c$>.
Proof. Let 0 = 0(x( ' ,(xj ')*) — 0 be a »-differential identity for a. Using
identity (9)' to bring * inside of derivations, we have 0 = 0(x(. ', (x*) ' ) — 0.
Suppressing derivation words for simplicity of notation, we write 0 = 0(x;, x* ).
Set xt = X),a,,'7,., where a; are fixed elements of a and rr are indetermi-
nates intended to range over C. Since * is of the first kind, x* = YL¡ annij ■
Substitute these x¡ and x* into 0 and write the resulting expression in the
following form:
^ £ aiñu • £ a*flu = £ bhlrfij ).
where b are C-linearly independent elements in the left Martindale quotient
ring of o and where cph(r]A) are differential polynomials in ir with coeffi-
cients also in C . If 17. are assigned values from C, then, since J2aijr¡n e °
and (£a/7i7;7)* =£«*-fyeff, 0(Eya,7V Eyflyfy) vanishes. Since ¿> are
C-linearly independent, cp,(r]A.) vanishes on C for each ¿>. Thus cph(t]A) = 0
is a differential identity (without *) for C. By the Kharchenko theorem,
cph(r]rk) = 0 is an identity on C , where rjjjk are new distinct indeterminates.
Since C is an infinite field, (ph{,lljk) is a trivial polynomial. Thus cph(t]A) also
vanishes trivially when r]tj are assigned values in O. So cp(Y^jajjr]jj, £y-a*y-i/,-7-)
vanishes when 17. range over <&. Since a- G o, »7.. G O are arbitrary and
since J2a, 1i ' wnere a; G er, 17. G O are typical elements of o ®c O, 0
vanishes on cr ®c. <P as desired.
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»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 275
As usual, each element in o ®c O can be interpreted as a linear map in
End^F) by defining
v(r ®a) = avr,
where v E V, r Eo , and a E O. Such a representation is faithful by Theorem
2.2 [7, p. 504]. Obviously, V is an irreducible o ®c <P module. It is easy and
well known that the commuting division ring of o®cQ> in End((J)F) is <P itself.
D is finite dimensional over C and hence must also be finite dimensional over
<P. For r E a, the range of r is finite dimensional over D and hence is also
finite dimensional over <P. So each element in a ®c <P is of finite rank in
End(0F). Thus o ®c <P coincides with its own socle.
The bilinear form ( , ) defined on D V is no longer a bilinear form on ^ V,
since the values of ( , ) fall in D but not in <P. However, by the Kaplansky
theorem, there exists a self-dual space U over a division ring A endowed with
involution such that
Su(U)Ca®cOCLu(U).
Since the socle of o ®c <P is nonzero, all the irreducible o ®c <P modules
are isomorphic and hence so are their commuting division rings. Thus we
may assume A = O. Since the two-sided Martindale quotient ring of a ®c <P
includes 7? as a subring naturally, by Proposition 4, it suffices to prove our Main
Theorem for o ®c <P. Replacing 7? by o ®c <P, we may assume henceforth
that the skew field of er coincides with its own extended centroid C.
Now are ready to give
Proof of the Main Theorem. It is well known that any ô E D + 7)int can be
uniquely extended to a derivation on the ring RF of left quotients. See for
example, Exercise 10 [6, p. 101]. The derivation thus extended will also be
denoted by ô. So we assume that any ô E D + D t is defined on the whole
RF.
Suppose that e is an idempotent in o. Then Ve is a finite-dimensional
subspace over C. Let us say that Ve is of dimension m over C. Choose
a basis {vx, ... ,v)n) for Ve over C. Also choose a basis {v( : a > m} for
V(l - e). Then {va: a > 1} is a basis of the whole space V . Let e„ E RF
(= A(V)) be the linear transformation such that v e „ — v„ and v e „ — 0v v y ' a aß p y (\¡i
for y / a . If a, ß < m , then et „ G eae . Also {e„ : a, ß < m} forms a basis
of eae over the field C.
For each S E D + 7>m , we define a new derivation ô on 7^.. (= A(V)) as
follows: Let a E RF be such that vna = J2ßcaßvß- Define ad E A(V) by
setting
vMs) = Hcißvß-
ß
Observe that e„ = 0. Also ô - ó acts trivially on C and hence must be
an inner derivation in 7?^.. Let r& E RF be such that ô = ô + ad(rs), where
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276 CHEN-LIAN CHUANG
ad(rs) is the inner derivation defined by rsE RF . Let M = {S: 8 G M) . For
ö", , ô2 G M, we define »5, < ô-, if and only if ôx < ô2. For A = oxô2-ôn,
where o¡ E M, define A = ¿),<57 • ■ • 8n. Note that if A is a regular word in M,
then A is also a regular word in M.
Suppose that 0 is a given »-differential identity of R. As in the proof of
the generalized Kharchenko theorem (see also the remark following Lemma
4), we may assume that 0 involves only one indeterminate x. Write 0 =
0(x ; ,(x ')*), where A are regular words in M. We may assume that A, >
A2 > • ■ • . For simplicity, let us also assume that {A.: j = 1,2, ...} is closed
under subwords.
Our strategy is to replace each ô E M in 0 by the more concrete derivation
â+ad(rs). But the difficulty is that * is defined only on Q and «oí on the whole
RF . The resulting expression after such substitution may involve (x )* and
r*. Even if x is assumed to range over Q, x may fall out of Q. Similarly,
rs may possibly not be in Q. In either case, the expressions (x )* and r*ö
make no sense.
Our solution is as follows: Using the identity (9), we bring 0(x ' ,(x ')*) to
0(x ' , (x*) '), where A* are regular words in M*. Observe the following: For
aEC and for ô E D + £>im ,
(S~) ,, *xc5n* / ¿,* àöl =((a ) ) = (a ) = a ,
since * is assumed to be of the first kind. Hence ô* - ô vanishes on C and
so does S* - 5 . Thus there exists ts E Rf such that ö* - ô + ad(ts).
Now, in cp(xAj, (x*)Aj ), we substitute ô + ad(rg) for ô in A, and ¿-r-aa7^)
for S* in A* respectively. Using identities (l)-(6), the resulting expres-
sion of such substitution can be brought to a new »-differential polynomial
y/(x ', (x*) '), where A. are regular words in M.
Define A, (/ > 1) inductively as follows: For 7 = 1, define A, = the highest
total degree of xA' and (xA|)* in 0. For j > 1 , define A, = the highest total
degree of x ; and (x J)* in those monomials of 0 whose total degree of x '
and (x ')* is A( for each i < j.
We define the leading part 0o(xA; ,(xAj)*) of 0(xAj ,(xA;)*) to be the sum
of those monomials of 0 whose total degree of x ' and (x ')* is A for each
j > 1. Similarly, we define the leading part y/0(x ' ,(x*) ') of y/(x ' ,(x*) ')
to be the sum of those monomials of y/ whose total degree of x ' and (x*) '
is A for each j > 1. It is important that the generalized polynomial 0o(zj. ,yß
coincides with the generalized polynomial y/0(z.,y ), where z., y. are new
indeterminates.
Set x — y.^. „s e «Ç „, where C tí are indeterminates intended to range¿—M <a ,/)<777 aß^ap ' ^aß °
over C and hence are assumed to commute with 7?,-. Since * is of the first
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»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 277
kind, we have x* = Ei<a>fi<m^C^ ■_ Suppose that A^= ô{ô2-■ ôsôs+x-■■
is a regular word in M, where Sx - ô2 = • ■ • = ös ^ ôs+x and 0 < s < p
(p = ch R). Since ea „ — 0 for ô E M, we have
A _ V—* „A
X ~~ /-*. eaß ^aß ■
\<a,ß<m
Using the formula ( 1 ) in the proof of Lemma 3,
Ä
(x*)A = £ e*«ßt«ß
\\<a,ß<m
= £ (<ߣß+s(<ßfCf- + ••■),l<a,/j<í?7
where the dots above denote a sum of terms (e*„) (Ç „) > where A , A <
<52<53 • • • . Substitute these expressions for x and (x*) into y/(x ', (x*) ')
and let t](Ç\) be the »-differential polynomial thus obtained.
We define the leading part »/0(í„«) to he the sum of those monomials of 77
whose total degree of Cri« (I < a,ß < m) is h ■ for each j > 1. Let Ç( „. be
new indeterminates. Observe that
%^t*ßj> ~ ^0 I ¿^ enß^nßj> Z-7 eaß*aßj\\<a,ß<m \<a,ß<m
— <rr)\ 2-7 eaßl*aßj> Z_j eaß^aßj
\j<rt,/l<777 \<a,ß<m
Now write r¡(C,L) = J2V vrl,X^aß) > where v E RF are C-linearly independent
and where r]v(Çni) are differential polynomials in Ç „ with coefficients in C.
When £„ are assigned values from C, r](Çi) assumes the constant value
0. Since v are assumed to be linearly C-independent, f],,(ClA) = 0 for each
v. Thus, for each v, %,(C„Jß) = 0 is a differential identity on C. By the
Kharchenko theorem, »/,,(£„«,■) = 0 is a generalized polynomial identity on C ,
where C„», are new indeterminates. Hence f/(£ „.) = 0 is also a generalized
identity on C (with coefficients in RF ). Since C is assumed to be infinite,
the leading part 10(Cltßß = 0 of r}(C,iißß = 0 is also a generalized identity on
C. Since
fu(£«/?./) = 00 I ¿^ eaß^aßj' Z_^ eaß^aßj\l<<!,/><777 I<«,/j<777
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278 CHEN-LIAN CHUANG
and since e„ (1 <a,ß<m) span eae over C , we obtain that cpQ(y ,y*) = 0
for yj e eoe .
Given any finitely many yx,y2, ... E o, there exists an idempotent e E a
such that yx,y2, ... G eoe . By the result of the previous paragraph, cp (y. ,y*)
- 0. Thus we have shown that 0o(y¡.y*) = 0 is a »-generalized identity on
a . By Proposition 4, 0O(;P ,y*) = 0 is also a »-generalized identity on Q .
Now consider
cp\xA', ixA'f) = 0(*ÂJ, (*V) - U*Aj - (*AT) •
Let 0q(x ', (x J)*) be the leading part of 0'(x ', (x ')*) • Repeating the above
argument, we can show similarly that cp'0iyj,y*) vanishes on Q. Continu-
ing in this manner, we can prove finally that the Main Theorem holds for
0(x\(xAO*).
Remark 1 . In our Main Theorem above, derivations are assumed to be in
Ant + C Der7?. However, as remarked in [4, p. 74], DerR may be replaced
by the larger set consisting of all derivations p E Der RF for which there exists
a nonzero ideal I of R such that IM ç R. Also the given differential identity
can be assumed to vanish only on a nonzero ideal of 7? instead of on the whole
ring R . The assertion of the Main Theorem above (together with its proof) still
holds for such a generalization.
Remark 2. As in [3] and [4], we have deliberately avoided the following two
fundamental questions: ( 1 ) With respect to a basis M of D linearly ordered by
<, does a differential polynomial give rise, via the basic identities (l)-(9), to a
unique differential polynomial of the form 0(x( ;, (*,J)*) > where A are distinct
regular derivation words in M (with respect to < ) and where 0(z(.., z*) is a
»-generalized polynomial? (2) Is the nontriviality of a »-differential polynomial
independent of the choice of M and the linear order < on it? Both questions
have affirmative answers. Actually, we can give a free-product treatment of *-
differential polynomials as initiated in [5], where all the basic notions can be
made precise and then the two questions above can be proved mathematically.
As such a treatment is rather long and does not seem to be immediately relevant
to our main theme here, we take up these matters somewhere else.
References
1. I. N. Herstein, Rings with involutions. Univ. of Chicago Press, Chicago, 1976.
2. N. Jacobson, Structure of rings. Amer. Math. Soc, Providence, R.I., 1964.
3. V. K. Kharchenko, Differential identities of prime rings, Algebra i Logika 17 (1978), 220-238.
4. _, Differntial identities of semipnme rings. Algebra i Logika 18 (1979), 86-119.
5. C. Lanski, Differential identities in prune rings with involution, Trans. Amer. Math. Soc. 291
(1985), 765-787.
6. J. Lambek, Lectures in rings and modules, Chelsea, 1976.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 279
7. W. S. Martindale III, Prime rings with involution and generalized polynomial identities, J.
Algebra 22 (1972), 502-516.
8. L. H. Rowen, Generalized polynomial identities, J. Algebra 34 (1975), 458-480.
9. _, Polynomial identities in ring theory, Academic Press, New York, 1980.
Department of Mathematics, National Taiwan University, Taipei, Taiwan 10764,
Republic of China
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