Dedicated to Professor Hisao Tominaga on his 60th birthday

RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie 11, Tomo X X X V (1986), pp. 349-363

ON RINGS WHOSE NUMBER OF CENTRALIZERS OF IDEALS IS FINITE

K. KISHIMOTO - J. KREMPA - A. NOWICKI

Let R be a ring. For the set F of all nonzero ideals of R , we introduce an equivalence relation in F as follows: For ideals I and o r, a r ~ J if and only

if VR(D = VR(or), where VR( ) is the centralizer in R . Let I R - F ~ ~ . Then we can see that n ( / R ) , the cardinality of I R , is 1 if and only f f , , ~ is either a prime ring or a commutative ring (Theorem 1.l). An ideal I of R is said to be a commutator ideal if I is generated by { s t - - t s ; s E ,S',t E T } for subset ,_q and T of R , and R is said to be a ring with ( N ) if any commutator ideal contains no nonzero nilpotent ideals. Then we have the following main theorem:

Let R be a ring with ( N ) . Then n(IR) is finite if and only if 12 is isomorphic to an irredundant subdirect sum of 8 f9 Z where 8 is a finite direct sum of non commutative prime rings and Z is a commutative ring (Theorem 2.1). Finally, we show that the existence of a ring R such that n ( a r R ) = m for any given natural number m .

Dedicated to Professor Hisao Tominaga on his 60th birthday

Introduction.

Let R be a ring with the center 6' and 8 a non empty subset of R.

By Vn(S) on 8' we denote the centralizer {x E R ; z s = sx for all s ~ S}

of 8 in R. For ideals I and or of R, we say S and or are equivalent if I ' = J'. Then the relation gives an equivalence relation of the set of all ideals of R. By In we denote the set of all equivalence classes of nonzero ideals of R and n(IR) denotes the cardinality o f IR. As is see in Theorem 1.1, a ring R has the property that n(IR)= 1 if and only if R is a commutative ring or a prime ring. Concerning this theorem, it seems to be interested to study the structure of a ring R such that n u n ) = n for a

350 K. KlStnMOTO - J. KREMI'A - A. NOWlCrd

given natural number n. Those studies are the purpose of this paper.

1. Prime rings.

As is known, if .R is a prime ring then V a ( / ) = (7 for any nonzero ideal I of R ([1], Lemma 1.1.6). First, we shall prove the converse is also valid if R is a non commutative ring, and so we can state the fol- lowing theorem.

THEOREM 1.1. R is a commutative ring or a prime ring if and only if n(IR)= 1.

Proof Assume R is a non commutative ring and n(IR)= I. Then I ' = 6' for any nonzero ideal I of R, and hence (7 contains the only zero ideal as an ideal of R. If R possesses a nonzero nilpotent ideal N such that N k-~ :/0 and N k = 0, then N (Z-_ VR(N k-t) = (7 yields a contradiction.

Thus R is a semi-prime ring. Let x be an arbitrary element of R - C . If :~I = 0 for some ideal 1, then I z = 0 since R is a semi-prime ring, and hence z ~ I ' . Consequently, I must be 0. Thus I d ~ t O if I and J are nonzero ideals.

For a nonzero ideal I of R, [/'1 denotes the equivalence class of I in IR. Since

VR( ~ ,13 = ns~it)Vn(I) = I',

Jcllq

[/1 contains the maximum element ~ J. We say the maximum element J~trl

is the maximum representative (abbreviate m-representative) of [I], and in what follows, we assume that [I] is the m-representative.

Definition. 1.2. Let [I](r [R]) ~ In.

(1) [I] is said to be a minimal class if I does not contain any m- representative J & / ' ) .

(2) [/'1 is said to be a maximal class if I is not contained in J for any m-representative J such that Y ~1I, R.

By m(IR) we denote the eardinality o f the minimal classes o f Ia.

LEMMA 1.3.(1) If Ia contains [M] such that M c C then m(/a) = 1 and [M] is the unique minimal class. If [M] is a minimal class, then

ON RINGS WHOSE ,NUMBER OF CENTRAI.IZFRS 351 I E [M] if a n d only i f I c_ M f o r a nonzero ideal I.

(2) I f [Mi] and [M2] are distinct minimal classes, then M t t~ M2 = O.

Proof. (1) Let [M] be a class such that M c 6'. Then VR(J + M ) = J' N M ' = J' N R = j I

for any m-representative J.

Thus M c: J. Let [M] be a minimal class.

which is contained in M, / ' _~ M'. If ! ~ [J], then

For a nonzero ideal I

VR(J + M ) = J' N M ' = I' n M ' = M ~

show that J + M _ M. Hence [J] must be [M] since J and M are m- representatives and [M] is a minimal class. Thus / ~ [M]. The converse is clear.

(2) If M1 n M2 ~r 0, we have a contradiction that M1 n M2 E [M1] and

M1 I~ M2 E [ME].

LEMMA 1.4. F o r any ideal I,

[ I ' , R I = {Ix, r] = xr - rx; ^{x ~} I', r ~ R}

is contained in the left (resp. right annihilator IR(I) (resp. rR(I)) o f I in R.

Proof. For any xr - rx E [ I ' , R] and a E I ,

( 2 r ~ r x ) a = x r a - - r I r a = t a x - - t a x = 0

show that [/',R] c: Is(I). By the same way, we can see that [I',R] c r R ( / ) . Let 8 and T be non empty subsets of R. By (8,23 we denote the ideal generated by [8,2"] and call it a commutator ideal. Concerning the commutator ideal, we put the following condition.

(N): Any commutator ideal contains no nonzero nilpotent ideal of R.

A ring which satisfies (N) is said to be a ring with (N). It is clear that a semi-prime ring and a direct sum of a semi-prime ring and a com- mutative ring are rings with (N).

352 x. vastn~tcrro ^- j. Z R r : M P A - A . N O W ~ O 0

2. Rings with (N).

In this section, we shall characterize the ring with (N) such that n(I~) is finite.

THEOREM 2.1 (1) A ring R is a ring with (N) such that n(IR) > I and m(IR)~10 if and only if R is isomorphic to an irredundant subdirect sum of a ring T = S t ~ g where $ is a non commutative semi-prime ring and g is a commutative ring or g = O.

(2) A ring R is a ring with (N) such that n(Ia) is finite if and only if R is isomorphic to an irredundant subdirect sum o f a ring T = S t ~ Z where $ is a finite direct sum o f non commutative prime ring and g is a commutative ring or g = O. More precisely,

(a) I f R is a ring with (N) such that n ( l n ) = 2 then $ is a non commutative prime ring and Z 7tO.

(b), (i) If R is a ring with (N) such that n(Ia) = n > 2 and m ( I a ) =

r a > 1, then S is a direct sum of m numbers o f non commutative prime

rings, Z = 0 and n = 2 "~ - 1.

(ii) i f R is a ring with (17) such that n ( I R ) = n > 2 and rn(IR) = 1 with the minimal class [M], then $ is a direct sum o f rn numbers o f non commutative prime rings where m = n(Itt) f o r R = R I M , g 7tO and n = 2".

To prove the theorem, we need several lemmas. In the rest of this section, we assume that the ring R is a ring with (27).

LEMMA 2.2. For any ideal I, la(/) = rn(I).

Proof. For any ideal I, r n ( I ) I is a nilpotent ideal and rR(l")I -- { E ( a / b i ^- biai); ai ErR(/), b, E I ) C_ (rR(I)I).

Thus r e ( / ) / - - 0 by (N). This means that r a ( / ) = IR(/).

By I* we denote la(/) for any ideal I.

LEMMA 2.3. (1) Any nilpotent ideal o f R is contained in C.

(2)

Proof. (1) Let N be a nilpotent ideal such that N ~ = 0. Then (N, R)k= 0 and this shows that ( N , R ) = O.

(2) Since ( R , R ) ~ [I] for some m-representative I, ( R , R ) c _ I and hence (I', R ) c : L But this yields (/',R)2 = 0 by Lemma 1.4. Consequefltly , we have ( / ' , R ) = 0 and hence VR((R,R))= I' = C .

ON RINGS WIIOSE NUMBER OF CE,WrRAIJZFRS 353 Concerning the uniqueness of a minimal class o f In, we have foUowing

LEMMA 2.4. Let n(IR) > 1 and I an m-representative.

(1) [1] is the unique minimal class if and only if I c C.

(2) I f I N I* vt 0 f o r some m-representative I, then re(In) = 1.

the

Proof. (1) Let [/] be the unique minimal class. Then wc have 1 , _ Y' for any m-representative J. Since (1", R) ~ [J] for some J,(J', R) is contained in Y. But this implies ( J ' , R ) 2 = 0 and hence J ' = C. Conversely, if I c_ C then [1] r [R] since n ( l n ) > 1. Thus the rest is clear by Lemma 1.3.(1).

(2) Since ( l n I * ) 2 = 0, INI*C_ C by Lemma 2.2.(1). Hence if In1* r IR contains the unique minimal class by (1).

LEMMA 2.5. Let I and J be ideals.

(1) 1" = {x ~ R; [x,R] c_ P } . (2) If 1" = Y" then 1"= Y'.

(3) If I and J are m-representatives, then 1' c_ J' implies J c I.

Proof (1) Let a E l ' . For any 9 and x E I ,

( a t - - r t l ) x = a r x - - t a x = r x a - - r x a = O.

Conve~ely, if [z,R] c_ I ~ then I x , 1 ] _ I n r . Since I n 1 " is a nilpotent ideal, we have ( x , / ) = 0 and x E 1".

(2) If x E / 1 , then [ x , R ] C _ I ' = Y ~ and so x E Y ' by (1). By the same way, we can see that y'c_ 1".

(3) Vn(I + J) = 1" N Y' = 1' show that I + Y c_ I and hence J _c I.

COROLLARY 2.6. Let I be an m-representative. Then I = I ~176 Further I" ~t O if t ~t R.

Proof. Since (I'~ = I*, Vn(P*)= Vn(I) by L e m m a 2.4.(2), and hence I ~ I. The rest is clear since I * ' = / ' .

LEMMA 2.7. Let [P] be a maximal class.

(1) P" = (P',R).

(2) P is a prime ideal.

Proof. (1) ( P ' , R ) _ P is clear. If ( P ' , R ) ~ t P then (P',R)c_Vn((P',R)')=

354 K. raslmaoa'o - J. ra~ElVtl,A - A. Nowlcra

= 6' since [P] is a maximal class. If ( P ' , R ) = 0 then we have a contra- diction (P] = [.,2] since P ' c : 6'. While, if ( P ' , R ) 3 0 , then there exists the unique minimal class [M] such that (/~R)___ M by Lemma 2.3. Hence ( P ' , R ) c_ M c: P and so ( P ' , R ) 2 = O. But this implies a contradiction P' = 6' again.

(2) Let I and J be ideals such that J ~ P and I Y c_ p. Then JP* ~tO since P * ~ P by Corollary 2.6, and I c.U_ ( J P ~ Since (JP')* 3- P, if ( J P * ) ' v t P, we have J P * C VR((JP*) ~ = C and so /~ has the unique minimal class [M] such that M 3_ JP*. Thus, 3"P* c_ M c_ P, and hence ( j p , ) Z = 0. But J P * = J(P~,R) is a subideal o f the commutator ideal (P',R). Hence we have a contradiction that J / ~ = 0. Thus I c_ (JP~ = P.

I ' * = I. The rest is clear since /'**--I.

LEMMA 2.8. Assume re(In)> 1.

(1) If [P] is a maximal class, then P* is an m-representative of a minimal class.

(2) If [M] is a minimal class, then M ~ is an m-representative of a minimal class.

Proof. (1) Let P* ~ [J]. I f [J] is not a minimal class, then there exists a minimal class [MI such that M c Y. Hence M ' ~ g'--VR(P*) and so M'~_ P+M*. Since [P] is a maximal class, we have either P + M ' E [R]

or P __. M*. If P 3- M ~ then P" c_ M ~176 = M, and this yields a contradiction P* ~ [M]. While, if P + M * ~ JR] then M c_c_VR(M') c _ V ~ ( P + M ' ) = C show that [M] is the unique minimal class by Lemma 2.3 and this is also a contradiction. Thus P* is contained in a minimal class [M]. Next, we shall show that P* is contained in a minimal class [M). Next, we shall show that P * - - M . Suppose that M ~_ P*. Then P A M 3 0 since P A M = 0 implies M c_C_ P*. Therefore Vs(P n M) = M ' ~_ P~. Thus M c_ P by Lemma 2.5.(3) and hence M* 3_ P*. Consequently, P" c_ M n M*.

Since M A M* is contained in r by L e m m a ,2.2, it must be 0 because of m(IR) > 1. Thus we have a contradiction P* = 0.

(2) Assume M * ~ [I1. Then we have M * = I by the same way as in the proof of (1). If there exists an m-representative J ~ t R such that M * g J , then M ~ J" by Corollary 2.5, and hence, M n J 3 0 . Consequently, we have M ' = V R ( M N J ) 3_ J ' , and so J ~ M by Lemma 2.5, (3). Thus we have a contradiction J* c_ M n M* = 0.

Combining Corollary 2.6 with Lemma 2.8, we can see that there ex-

ON RINGS WHOSE NUMBER OF CENTRAliZERS 355 ists a to one correspondence between the maximal classes and the minimal classes of IR by [P] --. [P*] if m(IR) > 1.

LEMMA 2.9. Let m(IR) > 1 and let {M,,;a E A) be all of the m- representatives of the minimal classes

o--E'o--E*'o.

t~EA ~ A

(2) Q E [R].

Proof. (1) For each t~, M , n Mg = 0 since re(In)> I. On the other hand, N,, c_ M~ by Lemma 1.3.(2), where Na = ~ M p . Thus M,, N N,, c_

#ca M , N M~ = O.

(2) If [Q] is a maximal class then Q*= M,, for some a by Lcmma 2.7.(1). Then we have a contradiction that M~ = 0. Further, if Q is contained in a maximal class [5'], then S * ~ Q* and this yields the samc contradiction. Thus Q E [R].

In the following, we shall study on In ,such that re(In)> 1.

LEMMA 2.10. Let In contain the unique minimal class [M]. Then (R, R) O M = 0.

Proof. Since M c_ (7, (R, R)M = 0 by Lemma 1.4. Hence Mg = 0 for Mo = (R, R) n M shows that Mo = 0.

LEMMA 2.11. Let In contain the unique minimal class [3/1] and let

=

R/M.

(1) /~ is a ring with (N).

(2) n(IR) is finite if n(In) is finite.

(3) m(Ik) > 1.

Proof. We may assume that each ideal of /~ is obtained by i for some ideal I of R such that I .3 M.

(1) If 1 is a nilpotent ideal which is contained in (So, To) for some subsetes So and To of /~, then I"C_ M for some integer n and I c_ (S, T)+

M = ( S , T ) ( g M where S = { s E R ; S E So} and T = { t E R ; ~ E T o } . Since ((S, T n / ) n c (8,T) n I " c_ (8,T) n M = O ,

356 K. K / S m M O T O - I. K R E M P A - A. N O W I C I a

we have (S, T) n I = 0, and hence I c_ M.

(2) If Vl~(/)= Va(/) is proved, we have V~(~ = Vl~(.]) if and only if V6(/) = V6(J). Hence, if I ~/M is an m-representative of 16, then i is an m-representative of I~. Conversely, for any m-representative of I~, it is obtained as i for some ideal I of R such that I ~ M. Further, on this I, V6(I)=VR(dO implies i _ ~ J, and hence, I _ 3 J + M ~ Y. Thus I is an m-representative of In. Consequently, [/] --, [i] gives a surjective map from I 6 - {[M]) to I~. Now, we shall show that V/K/)= VR(/).

v'~(h = {e e R; re,/] = 0) = {e e R; [ , , / ] _ M ) =

{~ e/~; [z,/] D [R, R] N M} = {~ E/~; [x,/] = 0} = V6(/).

(3) If It~ has the unique minimal class [K], then K" is contained in thc center of /~. By (2), Vh([t)= VR(R)= C, and hence, K is contained in C. Then, this yields a contradiction K c_ M.

Let T = ] ~ P ~ $ g where each / ~ is a non commutative prime ring

~IEA

and Z is a commutative ring or Z = 0.

Then T is a ring with (N). Under this notation, we have the follow- ing.

LEMMA

Proof. It is clear that A is a ring with (N). Let f be an injection from A to T and I~ be induced projection from A to Rot. For a subset

w = ((,~);,p e Rp, ~ e ^}

of T, W,, denote the set of all elements of (0, ..., 0, ra, 0,0,...) such that there exists an element (8#) in W with *a = to. Then, for any ideal I of A, f ( / ) , = 1(/) and f ( / ) , is an ideal of T. If ( z # ) E Vr(f(/)), then z~fo(a) = f , ( a ) z , ( a G I) implies :r, E Ca = C(P~), the center of R,,, if f(/).. :/0. Hence we may assume that

= E Cae E eZ,

ac'=At ~ A a

where Ax = {a ~ A; f(/')a 5i /

0)

f(/)# • 0).

oN RINGS WHOSE r,a:vm~ oF cr.mXAHzr~s 357 exists an m-representative I of A such that f ( / ) a ~t 0 for any a ~ A~ and f(/)0 = 0 for any ,8 ~ A2. Further, noting that each centralizer of an ideal T is obtained as

Eco ERooz,

aEAt aEA2

for some sisjoint union A~ t3 Au = A, we obtain n(/r) = n({Vr(I(I)); [_I"1 ~ IA)), and hence, the map

9b : (VT(f(I)); [ ~ E IA} "~ (V'I(A)(f(/)); [I] E IA}

such that Vr(](/))---, V/tA)(.t'(/)) gives a surjective map. Therefore, ~S is bijective if V r ( f ( / ) ) ~ t V r ( f ( J ) ) implies

VI(A)(I ( I") ) 5t Vl(A)(f ( d') ).

Now, we shall show it. Since V r ( f ( / ) ) C V r ( f ( J ) ) , there exists a e A such that

Vr(f(/'))~ = 6', and Vr(f(J))a =/2,, and hence VICA)(f(I))a = Ca n f(A),, and V1r = P~ 0 f(A)a. Further, / ~ O .t'(A)a is a nonzero ideal of f(A) since f(A) is an irredundant subdirect sum of T, and so, Ro n f(A)a 9~ Ca since it is also a nonzero ideal o f a non commutative prime ring R. If

VI(A)(f ( I") ) = 'Vl(A)(f ( f ) ) ,

the above implies a contradiction that R a n f(A)a = Ca n f(A)a.

NOW we are able to prove Theorem 2.1.

Proof of Theorem 2.1. (1) Let m(Ia) > 1 and let {Ma; a ~ A) be all of minimal classes of In. Then Q = ~ M a ~ JR] by L e m m a 2.8. Hence

a~_A

riM" c_ n V R ( 0 ) = C. By lemmas 2.7 and 2.8, each M,~ is a prime ideal, and r = 0 since it is contained in 6'. Further f"~M~ __ Ma. Thus

#Ca

(M';,,, c A) is an irredundant set of prime ideals, namely, R is isomorphic to an irredundant subdirect sum o f I' = ~ _ e R / M ~ . If one of R/M~ is

a E A

commutative, then M~ containes (R,R). But in this case, [M_'] becomes [R] since VR((R,R))= C by Lemma 2.3.

Next let m(IR)= 1 and [M] be the unique minimal class. Then, for the set of all maximal classes {Pa; oc ~ A) of R, r ~ Pa = M and ~ P~ c

a~_A #Ca

358 K . K I S H I M O T O - J. K R E M P A - A . N O W I C K I

M. Thus { P a , ( R , R ) ; a E A} is an irredundant set of ideals, and hence, R is isomorphic to

2" = e R / P a 9 R/(R, R).

otEA

If one of R[P,, is commutative, we have a same contradiction as in that of (1).

(2).(a). Let IR = {[M],[R]}. Then [M] is the unique minimal class and the maximal class. Thus M is a prime ideal by Lemma 2.7 and M N ( R , R ) = 0 by Lemma 2.10. Hence R is an irredundant subdirect sum of T = R I M (9 R / ( R , R).

(2).(b),(i) and (ii). These are direct consequences of (1) and lemma 2.12.

The following is an immediate consequence of Theorem 2.1.

COROLLARY 2.13. Let R be a ring with (N) such that n(IR) > 1. Then R is a semi-prime ring if and only if R satisfies one o f the following conditions (a) and (b):

(a) re(In) >_. 2.

Co) IR has the unique minimal class [M] and there exists a proper semi-prime ideal I such that I D__ (R, R) and I n M = O.

3. Rings without (N).

As is seen in Theorem 2.1, if a ring R is a ring with n(In)= n < oo such that , is neither 2'* nor 2 " - 1, then R is necessarily a ring without (N). It seems that there are various kind of rings of this type. In this section, we shall show few examples o f rings R without (/7) such that n(I~) is finite and caluculate n(Ia).

Let (o) be a cyclic group o f order , with a generator o and L a (o)-cyclic extension field over a field K. By use of L and o, we can construct a skew polynomial ring

i

whose multiplication is given by a X = Xo(a) for a ~ L. Then each ideal of S forms ( f ( X ) ) = f ( X ) S for a monic polynomial f ( X ) and f ( X ) is one

ON RINGS WHOSE NUMBER OF CENTRALIZERS

polynomial o f following types: X~g(X) w h e r e i = 0 , 1 , . . . , n - 1 and

359

g ( X ) = X "k + .x"(k-1)au_l + . . . + . X n a l + a o ( a . / e L)

(See [2], p. 38). H e n c e I = ( X ~) is at, ideal o f S. N o w w e consider a factor ring R = 8 / I . T h e n R = L O x L O . . . ~ z " - l L w h e r e z = X + I, z" = 0 and a~ = xo (a) for a ~ L. Further, {(1),(z), . . . , ( x " - 5 } is the set o f all nonzero ideals o f R. T h e n we have the following

THEOREM 3.1. R is a ring such that n(In)= , . Proof. We shall show that

Vn((xi)) = z"-I L 0 . .. ~) z"-~L 9 K .

For,

and

for any a E L . H e n c e we have

n-1

Ot = 2 :'iaJ E VR((xi)) (aj E L), j.o

n- l -i

etzia = ~ z~+Joi(aj) a, j.o

n-l-i

ziaot = E zi+Jui(a)aJ j=o

ai(a/)a = aja/(a).

Taking a E K , we can see that aj E L " , the fixed field o f g~, for j = O, 1 , . . . , n - i . Since there exists an element a E L such that gJ(a)~ta for 1 < j < n - l - i , aj must be 0 for 1 _ < j < n - l - i . Further, for i < n - 2 ,

aoz.l

and i f i = n - 1, ao = o"-~(ao) = ~,(ao) show that ao E L" = K. T h u s V R ( ( z i ) ) C_ z " - l L 0 . . . 9 z n - l L 9 K

and the c o n v e r s e is clear.

360 K , K I S t l / M O T O - J. K R E M P A - A . N O W l C K I

Let K be a commutative domain with the identity and M ( ( n , K ) the ring o f n x n matrices over K. By R we denote the ring of all upper triangular matrices over M ( n , K ) , that is,

R = {(aq G M ( n , If); aij = 0 for all i > j}.

Then we have the following.

Theorem 3.2. n(IR) = n 2 = 2n + 2.

To prove the theorem, we need several lemmas. For a family of subsets {I~j;i,j = 1 , 2 , . . . , n } of K, we put (Io.) = {(aijM(n,K);(i,33-entry aij o f (a~j) is contained in I o- for all i , j = 1 , 2 , . . . , n } . The following fact which is proved in ([3], Lemma 2.1) describes all ideals of R:

(a) I is an ideal o f R if and only if I = (/'ij) where lij are ideals of K such that Io' = 0 for i > j and

I~j + Ij~ C_ hh for i <_j < k.

Henceforth, for an ideal I of R, we denote I = (Iiy), where I i j ( i , j = 1 , 2 , . . . , n ) is an ideal of K which satisfies the above conditions.

In virtue of (a), we can easily see that the following LEMMA 3.3. Let I = (Iij) be an ideal o f R.

(1) I f I n = 0 f o r some p < q, then Iij = 0 f o r all i , j such that p < _ i < j < _ q .

(2) I f Ir~ ~l 0 f o r s o m e p < q, then I,, ~l 0 f o r alls, t such that s < p <

q~_t.

Proof. (1) Since I ~ + I / , c_ I n = 0 and I v j + I j ~ c__ I n = 0 , we have Ir~ = Ips = 0, and hence, I ~ + Iij c_ lpj yields Iij = O.

(2) Since I~, + I# c h t and I,q + Iqt C Iot, if there exists s < t such that I a = 0 we have I ~ = I,~ = 0, and hence, Ioz, + I n c:/,q yields a contradiction Ir~ = 0.

The following lemma describes the centralizer of an ideal of R.

LEMMA 3.4.

(hi)

Vn(/)= ~ Vn(e~) (p,q)~D(~)

ON RINGS WHOSE NUMBER OF C'.~.WrRALIZF~S 361 where e~ is the matrix unit whose (p, q)-entry is 1.

that

Proof. Let p < q and let I(p, q) = (I(p, q)q) be the subset of I such I@,q)q = / 0 if i r or j 4 p

[

Since Va(aer~) = VR(e~) for any a(:/0) ~ K and p < q, if (p, q) ~ D(u), then Vn(I'p, q)) = VR(e~), and hence,

= n = n I'7

(p,r (p,q)~D(~) ~,q)eDCu)

The followings are direct consequences of Lemma 3.4.

LEMMA 3.5.(1) For p <_ q,X = (aq) E VR(em) if and only if x n, = Xqq, xlp = 0 f o r i ~t p and x r = 0 for j ~t q.

(2) For p <_ q, VR(eaq) f3 Vn(et,,) 0 VR(e~) < VR(e~).

(3) G = {(zq) E R, ~-n -- z22 . . . x,~ and xq = 0 f o r i ~tj}.

LEMMA 3.6. Let I = (Iq) be an ideal of R. Then I E [R] if and only if In vl O or I,~, 40.

Proof. If In = I,,, = 0, then we can easily see that el,, ~ VR(/). Con- versely, assume that In :/0. Then, by Lemma 3.3, we have Iq =/0 for j = 2,3,...,n, and hence, by Lemma 3.4, VR(/)_c ['TVR(e~y). But Lemma

j=l

3.5. (1) shows that ~:)Vn(eO)= C. If Inn ~/0, we can prove the assertion

j=l

by the same way as in that of the case of fit vlO.

Let p and q be natural numbers such that 1 _< p < n and 1 < q <_ n.

For a nonzero ideal I = (I~) of R such that I r [R], we say that I is of type (p, q) if

I t p = I q , , = 0 , I 1 ~ 1 4 0 and

Iq-i,,r

If p is the smallest integer such that Itv = 0 and Itt~t ~'0, and if q is the largest integer such that Iq,,--0 and Iq_l,, ~'0, then

(b) I~, = 0 for any s > p + 1, and I,,,-7/0 for any s < q - 1 by Lemma 3.3.(2).

362 K . K I S H 1 M O T O - L K R E M P A - A . N O W I C K I

Thus, this pair @,q) determines uniquely. Further, it is worth to note that if I = (Io') is of type (p,c/), then

(c) I1j = 122 . . . liv--i = 0 and I,~ = I,,-1,, . . . Ir = 0 by Lemma 3.3.(1).

LEMMA 3.7. For any p and q such that 1 <_ p < n and 1 < q < n, there exists an ideal I o f type @,q). Conversely, any nonzero ideal I such that I f! [R]/s o f type @,q) f o r some p and q.

Proof. We put I = (hi) where Ilj = K for j = p + l , . . . , n , li,, = k for i = q - 1 , . . . , n and Iij = 0 for the others. Then I is an ideal by (a) and o f type (,v,q). Conversely, assume I = (Iij) is a nonzero ideal such that I r [R]. Then Ix~ = I,,,, = 0 by Lemma 3.6. If Ilj = 0 for j = 1,2, ...,n, then we have I~ s ---0 for all i , j by Lemma 3.3.(1), and hence I = 0. While, if /~,,=0 for q = 1,2,...,n, then we have also h i = 0 for all i , j . Thus I must be of type (p, q) for some p and q.

LEMMA 3.8. For any nonzero ideal I and J such that I , J r JR], VR(I-) = VR(J) if and only if I and J are o f same (p, q) type.

Proof. By Lemma 3.4, for any ideal I, Vs(1)= N (Vn(evq). Hence (n,~)~D(u)

if I and J are of some (p,q) type, then VR(/)= Vn(J). Conversely, assume that / is of (p,q) type, and J is of (p',g) type. Noting that

)

VR(]r) = N VR(•li) ("1 VR(ejn)

\ , - , 1 j and

VR(J) = ['7 V~(e~0 n VR(ei,)

\,=~+1 /

by (b) and (c), we have e ~ ~ VR(I)\VR(J) if p > p'. By the same way, we can prove VR(I) vlVR(J) if q > q'.

Now, we are able to prove Theorem 3.2.

Proof of Theorem 3.2. By Lemmas 3.7 and 3.8, we have

n(IR)=l+n({(p,q);

= 1 + ( n - 1) 2 = n 2 - 2 n + 2 .

ON RINGS WHOSE NUMBER OF CENTRAIJTa~S 363 REFERENCES

[1] Herstein I.N., Rings with involution, Chicago Lecture in Mathematics, The Univer- sity of Chicago Press, 1976.

[2] Jaeobson N., The theory of rings, Mathematical Survey, No. II, AaM.S. Providence, 1943.

[3] Nowicki A., Derivations of special subrings of matrix rings and regular graphs, Tsukuba J. Math.7 (1983), 381-297.

lC~.v~uto il 18 settembre 1985

Department of Mathematics Shinshu University N. Matsum~to 390

Japan Institute of Mathematics, University of Warsaw Warsaw 00-901, Poland and Institute of Mathematics, N. Copernicus University Torun 89-100, Poland