C L A S S I C A L R I N G S OF Q U O T I E N T S OF S E M I P R I M E P I - R I N G S

K. I. B e i d a r UDC 519.48

Throughout the paper, R is an associative ring with unity. Let X C_ R be a subset. Put

l(R; X) = {r ~ R i t Z -- 0}, r(R; X) - {r ~ RlXr : 0}.

Recall that a ring R is said to be a right pp-ring if, for any element a E R, there exists an idempotent v E R

such that r(R; a) : yR. Let n be a natural number. A ring R is said to be n-regular if, for any element

a E R, there exists an element b E R such that a~'ba "* = a'*. Below, a ring R is called strongly n-regular on the left if, for any element a E R, there exists an element b E R such that ba '~+1 : a "~. For definitions of

other notions, see [6, 12, 14].

It is well known that a commutative ring has a classical ring of quotients. In the class of noncommutative

rings the existence of a right classical ring of quotients is equivalent to the satisfaction of a right Ore

condition. Posner's theorem [11] shows that a prime PI-ring has a two-sided classical ring of quotients. On

the other hand, Sergman [5] and Markov [9] (independently) constructed examples of semiprime PI-r ings

without right classical rings of quotients. In 1979 Jondrup [8] showed that a semiprime finitely generated

algebra over a commutative ring, which is a right pp-ring with a polynomial identity, has a right ring of

quotients; at the end of the paper the author notes that he knows no example of a semiprime P / - r ing that

is a right pp-ring without a right classical ring of quotients. The main result of the present paper is stated

in the following:

T H E O R E M 1. Let R be a semiprime P/ - r ing that is a right pp-ring and let n be an upper bound of

indices of nilpotent elements of R. Then

1) R has a right classical ring of quotients Qez(R);

2) Qct(R) is strongly n-regular on the right and it is a right n-regular pp-ring with a polynomial identity;

3) a mapping P ~-, P N R gives a bijection of the set MaxSpec (Qcz(R)) of all maximal ideals of the

ring Qcz(R) and of the set MinSpec (R) of all minimal prime ideals of the ring R.

Also included in the paper (by courtesy of the author) is Markov's example of a semiprime PI- r ing that

is a right pp-ring without a left classical ring of quotients.

In relation to Theorem 1 it should be noted that a criterion for the existence of a right biregular classical

ring of quotients in a semiprime PI-r ing that is a right pp-ring was established in [4].

Let R be a ring, B(R) a Boolean ring of central idempotents of R, X(R) a set of maximal ideals of

B(R), Q(R) a maximal right ring of quotients of R, and O~(R) = RB(Q(R)) a subring of the ring Q(R)

generated by subrings R and B(Q(R)). Further, let z E X(R). Denote by R~ a localization of R with

respect to a multiplicatively closed subset B(R) \z , and by ~o~ : R --* R~, a canonical homomorphism of

rings. As is known, ~0z is a surjective homomorphism with a kernel zR. We will need the following lemma

plugged in the proof of Theorem 1 in [8].

Translated from Algebra i £ogika, Vol. 32, No. 1, pp. 3-16, January-February, 1993. Original article submitted October 27, 1992.

0002-5232/93/3201-0001 $12.50 (~) 1993 Plenum Publishing Corporation 1

L E M M A 1. Let R be a semiprime PI-ring that is a right/rp-ring. For an element r E Rx the following

conditions are equivalent:

1) r is a right regular element;

2) r is a regular element;

3) r = ~o~:(a) for some regular element a E R.

R e m a r k 1. Let R be a subring of a ring / / a n d let a E H. Suppose that H = RB(H). Then there

exist elements al , a 2 , . . . , a,, E R and orthogonal idempotents el, e2 , . . . , e,+ E B(H) such that a = ~ elai. i = l

Indeed, it is d e a r tha t a = ~ vjbj, where bl, b2,...,bm E R and vl, v2 , . . . , v , , E B(II). A finitely ] = 1

generated commuta t ive ring is Noetherian, as follows from Hilbert 's basis theorem; so a subring K =

(vl, v2 , . . . , v,~) of the Boolean ring B(H) , generated by elements vl, v 2 , . . . , v,,,, is a Noetherian Boolean

ring. Hence, it is finite. Let ex, e2 , . . . , eh be minimal idempotents of the ring K. Then vi = ~ ei, where i~s(j)

S(j) C {1, 2 , . . . , n}. Therefore, a = ~ eiai, where al = 2 { b j / i E S(j)}. i = 1

L E M M A 2. Let R be a semiprime PI - r ing that is a right pp-ring and let S = O~(R). Then S is a

right pp-ring.

P r o o f . Let q E Q(R). Put B = B(Q(R)), ( R : q ) a = {r E R / q r E R}. Further, let s = ~ r i e l , i = l

where ri E R, ei E B(R) for all i = 1 , 2 , . . . , n , and e~ej = 0 for i # j (see Remark 1 above). Clearly,

r ( R ; r i ) = viR, where vl = v l 2 E R. Put v = ~ e l v i + l - ~ ei. We showtha t r ( S ; s ) = vS. Suppose s t = 0 i=1 i = l

for some t E S. Let I = (R : t)R. Clearly, I is a dense ideal of R. Still further, we let d E I . Then td E R

and std = 0. Hence rleltd = s~dei = 0 for all i. Therefore, eitd(R : ei)R C r(R; ri) = viR. Consequently,

(1 - vl)eltd(R : ei)R = 0. Since (R : ei)R is a dense ideal of R, we have (1 - vl)eltd = 0 for all d E I . Hence

(1 - vl)eit = 0 for all i. Therefore,

vieit =eit, Y]~eivit = ~ e i t , i = l i = l

n n

V~ = E eivit + t -- E e l = t , i = l i = l

and v* = t. Consequently, r(S; s) C_ vS. The inverse inclusion is obvious. The l emma is thereby proved.

Let I be an ideal of a ring A. Put cA(I) = {a ~ A / a + I is a regular element of a factor ring A/I} .

L E M M A 3. Let R be a semiprime PI- r ing that is a right pp-ring. Let S = Oe(R) and z E X(S). Then

1) the ring S~ does not contain an infinite set of pairwise orthogonal idempotents;

2) S~ is a left and right pp-ring, in which each of the left (right) annihilators is generated by an

idempotent;

3) Sz satisfies the equality Cs+(O) = Cs.(fl(Sz)) , where fl(Sz) is a prime radical of Sz; 4) for any regular element c 6 S~, c[B(S~)] = B(S~);

5) S,/~(S=) is a semiprime Goldie ring that is a left and right pp-ring;

6) S~ has a right classical ring of quotients Q,~(S~:) that is a semiprimary PI - r ing ;

8) S has a right classical ring of quotients Oct(S);

9) [ Q o , ( s ) ] . =

P r o o f . Let Q -- Q(R) be a maximal right ring of quotients of the ring R and C a center of Q. From

Theorems 1 and 2 in [10] it follows that Q is a regular PI-f ing. Furthermore, Q= is a regular prime PI-ring

by Theorem 1 in [2]. By Posner's theorem [11], a prime Pl - r ing is a left and right Goldie ring. We know

that a right prime regular Goldie ring is a simple Artinian ring; hence Q= is simple Artinian. It then follows

from the inclusion S~ C__ Q= that S= does not contain infinite sets of orthogonal idempotents. By Lemma

2, S is a right pp-ring, and so Sx is also. Now, item 2) follows immediately from 1) and from Lemma 8.4

of [6]. It follows from Lemma 8.8 of [6] that the ring S~/~(S~) is a direct sum of finitely many prime pp-rings.

By Posner's theorem, it is a semiprime Goldie ring. By Lemma 1, each fight non-zero divisor of S= is a

non-zero divisor. Therefore, statements 3) and 4) follow from Theorem 8.10 in [6]; 6) follows from Theorem

8.11 in [6]; and 7) follows immediately from Corollary 1 of [1] because S~ is a PI-r ing.

It follows from our Lemma 1 that ~o~(Cs(O)) = Cs.(O). Since the ring S~ satisfies the right Ore

condition for all z • X ( S ) , the ring S also satisfies this condition (see Theorem 1 in [8]). Therefore, the

ring S has a right classical ring of quotients. The equality [Qc~(S)]~ = Qcz(S~) is implied by the equality

= v s . ( o ) .

L E M M A 4. Let A be a left and right pp-ring such that each right regular element is regular and the

factor ring A/ f t (A) has a finite Goldie dimension oa the right. Then A has a right classical ring of quotients

Q - Qd(A) and contains orthogonal idempotents vl, v2 , . . . , vm such that

1) vl +v2 + " ' + v , ~ = 1;

2) viAv i = 0 for i > J;

3) viAv c_ Z(A) for i < i; 4) v~Avi is a prime Goldie ring for all i = 1, 2 . . . . , m;

5) viQvi = Q~z(viAvi) for all i = 1 , 2 , . . . , m .

P r o o f . It follows from Theorem 8.11 in [6] that the ring A has a right classical ring of quotients

that is semiprimary. Further, Theorem 8.10 in [6] entails CA(O) = CA(~(A)) and cft(A) = fl(A) for all

c • CA(O). Since the factor ring A/f l (A) has a finite Goldie dimension on the right, A does not contain

infinite sets of orthogonal idempotents. Lemma 8.8 in [6] implies the existence of orthogonal idempotents m

Vl, v2 , . . . , v,~ • A satisfying conditions 1)-4). Clearly, a/ft(A) = (~ viAvl. i----1

Further, each element of the ring Q is representable in the form dc - I , where d • A and c • CA(O).

By the above, we have CA(O) = CA(,G(A)). Therefore, c = a + b, where b • f~(A) and a • ~ viAvi is a i = 1

non-zero divisor of the ring (~ viAvi. Note that a is also a non-zero divisor of the ring A. Consequently, i = 1

aft(A) = fl(A) and b = at for some t • fl(A). So, c = a + b = a(1 + t) and dc -1 = [d(1 + t ) - l ] c -1. Clearly,

r = d(1 + t) -1 • A. Since a • ~ v~Avi, we have a = v~avi + (1 - vi)a(1 - vi). Therefore, i = 1

a-lviavi -= [a-:{viavl + (1 - vi)a(1 - vi)}]vi -- (a-la)vl -= vi

and v i r a - l v i v i a v i = ( v i r ) ( a - l v i a v i ) = virvi • viAvi. Now, it is clear that viAvi C viQvi C Qcl(viAvi).

Consequently, viQvl is a prime semiprimary ring. Hence, viQvi is a simple Artinian ring. Since viAvi C

viQvi, we have viQvi = Qel(viAvl).

L E M M A 5. Let R be a semipfime PI-ring that is a right/rp-ring; let S = O~(R), Q = Qct(S) be a

fight classical ring of quotients of the ring S, and J an ideal of Q. Then, for any regular element c of S, we

have

(c + s n J) (s/s n J) = / s ( s / s n :) .

P r o o f . Let ¢ : Q --, Q / J be a canonical h o m o m o r p h i s m of rings. P u t O -- ¢ ( S ) and N - f l ( S / S N J) =

f l (D) . Clearly, Q / J is a right classical ring of quotients of the ring D with respec t to a mul t ip l ica t ively

closed subset of non-zero divisors ¢ ( C s ( 0 ) ) . Since D is a P I - r i ng , we have f l (Q/L) = N[Q/J] (see Corol lary

1 in [1]). In par t icu la r , N C ~ ( Q / J ) . Let • E X ( S ) and ~0: Q ~ Q~, r : Q / J --* Q / ( J + zQ) = Q~/]~,

and ~ : qx = Q/zQ --* Q/(I + ~q) be canonical homomorph i sms of rings. Clearly, ~ = 7"¢. I t follows

f rom L e m m a 3 t ha t Qx is a semipr imary ring. The radical of a homomorph ic image of a s e m i p r i m a r y ring

is equal to a h o m o m o r p h i c image of the radical. Therefore,

= + c_ c_ ,-(N).

I t follows f rom L e m m a s 1 and 3 tha t the conditions of L e m m a 4 hold for the r ing S~. Let vl , v 2 , . . . , vm E

S~ be o r thogona l i dempo ten t s with the propert ies listed in i tems 1) th rough 5) in L e m m a 4 and let r E

S~. Clearly, r - ~ v, rvj. Further , Q~ - Qcz(S~) and viQ~v, = Qcl(v,S~v,) (see L e m m a s 3 and 4

above) . Therefore , the equal i ty ~(fl(Q~)) = f l (Q / ( J + zQ)) entails cr(r) E f l ( Q / ( J + zQ) ) if and only if

= 0. H e n c e , e + impl ies = I n a s m u c h as e i=i i<j {<j

the re la t ion ~(r) E f l ( Q / ( J q- xQ)) entails a ( r ) : q( t) for some t E fl(S=:).

Let a E N . Clearly, a e ¢(b) for some b E S and o'~o(b) : r ( ¢ ( b ) ) = r(a) E r ( N ) C_ f l (Q / ( J + xQ)). By

the above, there exists an e lement s E S such tha t !o(s) E fl(Sx) and a[~(s)] = crib(b)] --- ~'(a). Obviously,

~(c) is a non-zero divisor of the ring S , . Hence ~o(c)fl(S~) = f l (S , ) and ~o(s) = ~o(c)~o(t,), for some t , E S.

Wi thou t loss of genera l i ty we m a y assume s = c t , . Clearly, ~o(b - ct~) = a~o( b ) - (r~(s) : r(a) - r(a) = 0

and b - c t ~ E J + z Q . Fur ther , xQ = {q E Q I there exists an element e E B ( S ) \ x such t h a t eq = 0}.

Therefore, e,b - ce,t~ E J for some e , E B ( S ) \ z . Since the topological space X ( S ) is compac t , we

conclude t h a t b - ¢t E J for some ~ E S. So, a = ¢(b) = ¢ ( c ) ¢ ( t ) . Still fur ther , c is an invert ible

e lement of the ring Q. So, ¢(c) is an invertible e lement of the ring Q / J . By the above, a E g C f l (Q/J ) .

Therefore, ¢ ( t ) E ¢ ( c ) - ~ a E f l (Q/J ) . Inasmuch as f l (V) n U c_ 13(V) holds for all rings U C_ V, we have

¢( t ) C_ f l(D) : N . Consequent ly , ¢ ( c ) N = N. The l e m m a is thus proved.

Let a E R. I f a is a n i lpotent element, we put In (a) = m i n { n i a ~ = 0}. Otherwise, pu t In (a) = oo.

Fur ther , set In (R) = sup{In (a)l~ ~ R}.

L E M M A 6. Let R be a semipr ime P I - r i ng , n = I n ( R ) < oo, a ~ R, and r ( R ; a n ) -= vR, where

v -- v ~ E R. T h e n a ~' q- v is a regular element for any k > 1.

P r o o f . Let z ~ r (R ; a ~' T v). Then

(a I: + v)z = O. (1)

Let us mul t ip ly equal i ty (1) on the left by a" . Since a~v = O, we have a"+~z = 0. But r ( R ; a '~) = r ( R ; a r*+~)

for all k >_ 0, as follows f rom L e m m a 5 in [3]. Therefore, z G r (R;a ~) -= yR. Thus , vz - z. Mult ip ly ing

(1) on the left by a " - ~ , we obta in 0 -= a'*-~vz = a"-~z and z G r (R; a" -~ ' ) . Mul t ip ly ing (1) on the left

by a '~ -~ ' , we get 0 = an-2kvz -= a ~ - 2 k z and z E r ( R ; a ' * - ~ ' ) . The process m a y persis t unti l n - ~k > k.

Ult imately, we get z ~ r (R ; a'), where s _< k. But z ~ r (R ; a ~') in any case. Hence 0 = (a ~ + v)z = vz = z.

Consequently, a ~ + v is a right regular element. But a right regular e lement in a semipr lme P I - r i n g is

regular, as is impl ied by T h e o r e m 3 in [7].

L E M M A 7. Let R be a semipr ime ring, Q = Q(R) its m a x i m a l right ring of quot ients , and R C_ H C_ Q

an a rb i t r a ry subring. Let a e R and r (R; a) = vR, where v = v = • R. T h e n r ( H ; a) = vH.

P r o o f . Clearly, r(H; a) D vH. Let q E r (H; a) and I = (R : q)R- Obviously, I is a right dense ideal of

R and qr E r (R; a) = v R for all r E I . Therefore, (1 - v)qI = 0 and (1 - v)q = 0. Consequently, q E v H

and r (H ; a) = v H .

L E M M A 8. Let A be a right pp-ring and H D A a ring such that H = A B ( H ) and r ( H ; b) = r(A; b)H

for all b E A. Further, let a E A. Then aH (1 A = aA.

P r o o f . Let z = ah E aH n A. By Remark 1, the element h is representable in the form h = ~ ziei, i = l

where zi E A, ei G B ( H ) , and eiej = 0 with i # j . The proof will proceed by induction on n.

Suppose n = 1. Then z - - azle l E A. By assumption, r ( A ; a z l - z ) = v i A for some vl = vx 2 E A.

M o r e o v e r , , ( H ; = = v H. S i n c e * e l = ( zlel)el = = * , w e h a v e = 0

and ex E r(H; azl - z) = v lH . Hence vxel = el. We have

Thus, z = aZlV I.

We turn to the general case. By assumption, r(A; a z 1 - - z ) = v i A for some Vl = vl 2 q A. Since

zel = a ( E z ,e , )e l = az le l , we have (azl - z)el - 0 and el G r ( H ; a z l - z) = v l H . Hence vle l = el and i = l

el(1 - vl) = 0. We have

z v l : azlv a n d z ( 1 - = z e,)(1 - --- a[ z (1 - vl)e ]. i = 1 i = 2

By the induction hypothesis, z(1 - vl) = ad for some d E A. So, z ---- zvx + z(1 - vx) = a(z l ,~ + d). The

lcmma is thus proved.

L E M M A 9. Let R be a PI - r ing and H a right classical ring of quotients of R with respect to a

multiplicatively closed subset T C C•(0). Then the mapping P ~-* P n R gives a bijcction of the sets

MinSpec (H) and MinSpec (R), in which case P = (P N R)H.

P r o o f . I t follows from Theorem 1 in [1] that H is a P/-r ing. Let K be a prime ideal of H and

¢ : H -* H / K a canonical homomorphism. Clearly, the ring H / K is a right classical ring of quotients

of the ring ¢ (R) with respect to a multiplicatively closed subset ¢ (T) . Since H / K is a prime PI- r ing ,

Corollary 1 of [1] entails that ¢ (R) is a semiprime ring. Further, each central element of ¢ (R) is a central

element of the ring H / K , whereas a center of H / K is an integral domain. Now, making use of Theorem 2

from [3], we state that ¢ (R) is a prime ring. Consequently, K n R is a prime ideal of R.

Let P be a minimal prime ideal of the ring R and let U = R \ P . Choose an ideal K of the ring H to

be maximal relative to the property K n U = 0. Obviously, K is a prime ideal and K n R C_ P. By the

above, K n R is a prime ideal of R. Since P is a minimal prime ideal, we have K n R = P . Consequently,

L = H P H C_ K and L n R - P . Hence, H / L is a right classical ring of quotients of the prime ring R / P

with respect to a multiplieatively closed subset ( T + L ) / L . This implies that H / L is a prime ring and L is a

prime ideal of H. Obviously, /~ is a minimal prime ideal of H. Since H is a right classical ring of quotients

with respect to T, we have L = (L n R ) H = P H .

Let K be a minimal prime ideal of the ring H and let P = K n R. Tha t P is a pr ime ideal of R follows

from the above. Suppose that the prime ideal P is not minimal. Then it contains some minimal prime ideal

M of R. Our argument above implies that H M H is a minimal prime ideal of H and H M H n R = M.

Since M C P C_ K, we have H M H C K. We are led to a contradiction with the minimali ty of the prime

ideal K . Thus, the rule K ~-~ K N R induces a bijection of the sets MinSpec (H) and MinSpec (R).

P r o o f of Theorem 1. Let Q be a right classical ring of quotients of the ring S = O~(R), a, s E R, and

s E CR(0). Suppose that at :/: sb for all t E CR(0) and b E R. Let at = sb for some t E CR(0), b E S. It

then follows from our Lemmas 7 and 8 that sb = sd for some d E R, a contradiction. We can assume

at # sb for all t E CR(0) and b E S. (2)

Denote by U(Q) a group of invertible elements of the ring Q. Choose an ideal J of Q to be maximal with

respect to the property

a t - sb~ J for all t ~ CR(0) and b ~ S. (3)

The existence of a certain (e.g., zero) ideal with the above property (3) is guaranteed by (2).

Letting ¢ : Q -~ Q / J and D = ¢(S) , we show that

= ¢ ( c R ( 0 ) ) c c D ( 0 ) = ¢ ( C s ( 0 ) ) c (4)

Clearly, Ca(O) C_ Cs(O) (see, e.g., Lemma 7 above). Therefore, CR(0) _C Cs(O) C U(Q) and ¢(CR(O)) C

¢(Cs(0)) C U ( Q / J ) . Consequently, C¢(R)(0) _D ¢(CR(0)). Let ¢(a) E C¢(.~)(0) and r ( R ; a '~) = yR. By

virtue of Lemma 6, a + v E CR(0). So, ¢(a + v) E ¢(CR(0)) C C¢(R)(0 ). Since ¢(a)n¢(v ) = 0 and

¢(a) e C¢(R)(0), we have ¢(v) = 0. Hence ¢(a) = psi(a + v) E ¢(CR(0)) and C¢(R)(0) = ¢(CR(0)). That

CD(O) = ¢(Cs(0) ) can be proved in a similar manner.

Suppose N = f l ( Q / J ) • O. Let M = ¢-1 . Clearly, M D J . By the choice of ] , we get at - sb e M for

some t E CR(0) and b @ S. Therefore, ¢(a)¢( t ) - ¢(s)¢(b) E N = ¢ (M) . Hence ¢ (a )¢ ( t ) - ¢ ( s )¢ ( t ) =

d E N N D C f l (D). From our Lemma 5 it follows that d = ¢( s )¢ ( r ) for some r E S. Therefore,

¢ (a )¢ ( t ) - ¢( s )¢ (b - r) = 0 and at - s(b - r) E ], which contradicts the choice of J . This means that

Q / J is a semiprime ring. Further, it follows from relation (4) that Q / J = Qc~(D). Since D is a PI-ring,

we have f l (D) = 0 (see Corollary 1 in [1]).

Thus, D is a semiprime ring. Since D = ¢ ( R ) B ( D ) , the ring ¢(R) is also semiprime. Consider the case

where the ring ¢(R) is prime and, hence, is a Goldie ring [11]. Now, relation (4) implies that a t - sb E J

for some t E CR(0) and b E R, a contradiction.

It remains to consider the case where ¢(R) is a semiprime ring that is not prime. By Theorem 2 of [13],

¢(R) contains a non-zero central element ¢ ( f ) that is a zero divisor. Clearly, ¢ ( / ) is a central element of

the ring D. Let r(R; f " ) = vR, where v = v 2 e R. Lemma 6 entails that f " + v E C;~(0). In view of (4),

¢(f'~ + v ) E CD(O). Let ¢(z) E r (D;¢( f '* ) ) . Since f " v = 0, we have

¢ ( f - + - = ¢ ( / - ( 1 - = = = 0.

Therefore, ¢((1 - v)z) = 0, ¢(z) = ¢(v)¢(z) , and r(D; ¢(f '*)) = ¢(v)D. Since ¢(f '*) is a central element,

¢ ( v ) D is an ideal of the semiprime ring D. Hence, ¢(v) is a left identity element of the semiprime ring

¢(v)D. Therefore, ¢(v) is an identity element of the ring ¢(v)D. Consequently, ¢(v) is a central idempotent

of D. It follows from the above that Q / J = Qd(D). Therefore, ¢(v) is a central idempotent of the ring

Q/J . Clearly, ¢(v) :~ 0, 1.

By the choice of J , there exist elements Q, t2 E CR(0) and bl, b~. E S such that aQ - sbt E J + Qv and

at2 - sb2 e J + Q(1 - v). Therefore, a[t~(1 - ~) + ~.v] - ~[bl (1 - v) + b2v] E J . Clea~ly, ¢([t1(1 - v) + t~.v]) =

¢(t l )[1 - ¢(v)] + ¢( t2)¢(v) . Since ¢(Q), ¢(t2) E ¢(CR(0)) : C¢(R)(0) and ¢(v) is a central idempotent of

D, we conclude that ¢ ( t l ) [ 1 - ¢ ( v ) ] + ¢ ( t 2 ) ¢ ( v ) E C~(R)(0). Consequently, there exists an element t E CR(0)

such that ¢( t ) = ¢([Q(1 - v) + t~v]). Hence t - [tl(1 - v) + t~v] E J and at - s[bl(1 - v) + b2v] E J. Tiffs

conflicts with the choice of J . The ring R satisfies the right Ore condition and has a right classical ring of

quotients H - Qcz(R). It follows from Theorem 1 in [1] that H is a PI-r ing. Let P E MinSpec (R). By

Lemma 9, P ---- g • R for some g 6 MinSpec (H). Theorem 2 in [3] implies In ( R / P ) <_ In (R). Since a

classical ring of quotients of a prime PI-r ing is a localbation of this ring with respect to a multiplicatively

closed subset of central non-zero divisors [13], we state that In ( H / K ) = In ( R / P ) <_ In (R). A semiprime

ring is a subdizect product of factor rings relative to minimal prime ideals. Therefore, In (H) = In (R).

Let ac - x E H , where a 6 R and c 6 C/z(0). By assumption, r (R;a ) ---- vR , where v ---- v ~- E R. Clearly,

r ( H ; a c -1) = c r ( H ; a ) = c v H = c v c - l H (see Lemma 7 above). Therefore, H is a right pp-ring.

Let a E H. It follows from the above that r (H; a'*) -- v H for some v -- v ~ 6 H. Lemma 6 entails

that a '~ ÷ v is a non-zero divisor of the ring H. Since H = Qcz(R), a '~ + v is an invertible element

of H. Clearly, a'~(a '* ÷ v) - x - 1 - v(a '~ ÷ v) -x . Multiplying this equality on the left by a'*, we get

a2'*(a'~+v) - z -- a'*. Consequently, H is n-regular on the right. Since ( a " + v ) v - v, we have ( a " . ÷ v ) - t v = v.

Therefore, (a '~ .4- v ) - t a '~ -- 1 - (a '~ .4- v ) - l v = 1 - v. Multiplying this equality on the left by a", we obtain

a'~(a '* .4- v ) - l a "* - a '~. Hence, H is n-regular.

A prime n-regular PI- r ing is simple Artinian. So, MinSpec (H) -~ MaxSpec (H) . Now, our theorem

follows readily from Lemma 9.

E x a m p l e 1. Let F be a field, F[z] a polynomial ring in variable z over F , and M2(F[z]) a ring oo oo

of 2 × 2-matrices over the ring F[z]. Put R~ = M~(F[z]) for all i, R ---- 1-[ R4, D --- ~) R~ C_ R, and i = 1 i = 1

epq -= "[Eipq}~=l E R, where { ~ q ]1 < p, q < 2} is the system of matr ix units for Ri. Further, put

K = e l iF[z] -4- e21F[z] -4- e22F C_ R. Clearly, K is a subring and D is an ideal of R. Therefore, H = K -4- D

is a subring of R. Clearly, H is a semiprime PI-r ing. We will show that H is a right pp-ring, and that

the left Ore condition fails in H. Obviously, CK(O) = "[eixf .4- e2zg -4- e22h] f , h # O, f , g E F[z], and

h E F}. Put a = e2i and c = e i i z + e 2 2 . Suppose that b a = d c f o r some b E CK(O) and d E K. Let

b = e i i f + eT.ig + e22h and d = e i i u + e2iv + e22w. Then h -=- zv, where h E F, v E F[z]. We are led to

a contradiction. Clearly, H / D = K , CK(O) C_ CH(O), and (CH(O) + D ) / D = CK(0). Therefore, ba # d c

for all b E CH(O) and d E H. Hence, the ring H does not satisfy the left Ore condition. Since F[z] is a

ring of principal ideals, M2(F[z]) is a hereditary ring and, in particular, a right pp-ring. Then H is a right

pp-ring if K is a right pp-ring. It remains to observe that r (K; e ~ u + e21v -4- e22w) = 0 if u # 0 and v # 0;

r (K; e l i u + e2~v) = e22K if either u # 0 or v ¢ 0; r (K; e~.iv + e22w) = (e l l - e 2 i v w - i ) K if w # 0.

I would like to express my gratitude to A. Mikhalev for encouragement, and to R. Wisbauer and V.

Markov for helpful criticism.

R E F E R E N C E S

1. K. I. Beidar, "Classical rings of quotients of PI-algebras," Usp. Mat. Nauk, 33, No. 6, 197-198

(1978).

2. K. I. Beidar, "Rings of quotients of semiprime rings," Vestn. MGU, No. 5, 36-43 (1978).

3. K. I. Beidar and A. V. Mikhalev, "Semiprime rings with bounded indices of nflpotent elements," Tr.

Sere. ira. I. G. Pe~rovskogo, No. 13, 237-249 (1988).

4. K. I. Beidar and R. Wisbauer, "Strongly semiprime modules and rings," to appear in Usp. Ma~.

Nauk.

5. G. M. Bergman, "Some examples in P/ - r ing theory," Israel J. Math. , No. 18, 257-277 (1974).

6. A. W. Chatters and C. R. Hajarnavis, Rings with Chain Conditions, Pitman, Boston (1980).

7. L N. Herstein and L. W. Small, "Regular elements in PI-rings," Pac. Y. Math., 36, No. 2, 327-330

(1971).

8. S. Jondrup, "Rings of quotients of some semiprime PI-tings," Commun. Algebra, 7, No. 3, 279-286

(1979).

9. V. T. Markov, "Maximal rings of quotients for uncancellable subdirect products," Mat. Issled., 9,

No. 2, 237-345 (1974).

10. W. S. Martindale, "On semiprime PI-rings," Proc. Am. Math. Soc., 40, No. 2, 365-369 (1972).

11. E. Posset , "Prime tings satisfying a polynomial identity," Proc. Am. Math. Soc., 11, 180-184 (1960).

12. C. Procesi, Rings with Polynomial Identities, Marcel Dekker, New York (1973).

13. L. H. Romen, "Some results on the center of a ring with a polynomial identity," Bull. Am. Math. Soc., 79, 219-223 (1973).

14. B. Stenstrem, "Rings and modules of quotients," Leer. Notes Math., 237 (1971).

Translated by the author

S E M I S I M P L E P E R I O D I C G R O U P S OF F I N I T A R Y T R A N S F O R M A T I O N S

V. V. B e l y a e v UDC 512.544

I N T R O D U C T I O N

In the present paper we continue the research of periodic groups of finitary transformations initiated

in [1, 2]. Recall that a linear transformation g of a vector space V is called finitary if g acts identically

on some subspace of finite dimension in V. A maximal locally solvable normal subgroup in a locally finite

group G is called a locally solvable radical and is denoted by S(G). If S(G) ---- 1, a nontrivial group G is

said to be semisimple. The main goal of the present paper is to prove the following two theorems.

T H E O R E M A. Let G be a group of finitary transformations and N a semisimple periodic subgroup

normal in G. Then N contains minimal normal subgroups in G and for any one of these subgroups M one

of the following is valid:

1) M is a simple group;

2) M is representable as a direct product of finite simple nonabehan groups;

3) M is representable as a direct product of infinite simple Lie type groups of finite rank.

T H E O R E M B. Let G be an infinite simple periodic group of finitary transformations on a space over

a field of characteristic p. Then the following are valid:

1) if p -- 0, then, for each finite subgroup K of G, there exists a finite quasisimple subgroup H that

contains K and is such that K N Z(H) = 1;

Translated from Algebra i Logika, Vol. 32, No. 1, pp. 17-33, January-February, 1993. Original article submitted May 25, 1992.

8 0002-5232/93/3201-0008 $12.50 © 1993 Plenum Publishing Corporation