rings which are essentially supernilpotent

This article was downloaded by: ["University at Buffalo Libraries"]On: 02 May 2013, At: 00:11Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

Rings which are essentially supernilpotentGary F. Birkenmeier aa Department of Mathematics, University of Southwestern Louisiana, Lafayette, 70504Published online: 27 Jun 2007.

To cite this article: Gary F. Birkenmeier (1994): Rings which are essentially supernilpotent, Communications in Algebra,22:3, 1063-1082

To link to this article: http://dx.doi.org/10.1080/00927879408824893

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form toanyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss, actions,claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.

http://www.tandfonline.com/loi/lagb20

http://dx.doi.org/10.1080/00927879408824893

http://www.tandfonline.com/page/terms-and-conditions

COMMUNICATIONS IN ALGEBRA, 22(3), 1063- 1082 (1994)

RINGS WHICH ARE ESSENTIALLY SUPERNILPOTENT

Gary F. Birkenmeier

Department of Mathematics University of Southwestern Louisiana

Lafayette'Louisiana 70504

INTRODUCTION

In [8] Fisher defines an ideal L of a ring R to be essentially nilpotent if it

contains a nilpotent ideal N of R which is essential in L (i.e., N has nonzero

intersection with each nonzero ideal of R which is contained in L). He then

shows that the prime radical of an arbitrary ring is essentially nilpotent. In

[I] Eslami and Stewart generalize Fisher's work by showing that each ring

contains a unique largest essentially nilpotent ideal, E N ( R ) . They investigate

the basic properties of this ideal and consider its behavior in related rings.

The purpose of this paper is to extend the concept of essential nilpotency to

a theory of essential supernilpotency. We will consider rings and ideals which

are essential extensions of a supernilpotent radical. Our results will encompass

those of Fisher and Eslami and Stewart and provide a general framework on

which we can base further investigations. For example in a sequel [&] to this

paper several well known "splitting theorems" are shown to be special cases

of our theory.

Copyright 1994 by Marcel Dekker, Inc.

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3

1064 BIRKENMEIER

Throughout this paper all rings are associative but not necessarily with

unity. The word "ideal" will mean a two-sided ideal unless it is preceded by

either of the words "left" or "right". R will always denote a ring, and I a R

means that I is an ideal of R. For X R, < X >R, rR(X), lR(X) denote

the ideal of R generated by X , the right annihilator of X in R, and the left

annihilator of X in R, respectively (the subscript R may be deleted when

the context is clear). Let X Y C_ R, then we say a subset X is (right

essential) R-essential in a subset Y [equivalently, Y is a (right essential) R-

essential extension of X] if X has nonzero and nonempty intersection with

every (right) ideal of R which has nonzero and nonempty intersection with

Y. Left essentiality is defined analogously. When X and Y are right ideals

our definition of right essential agrees with the usual definition. We say X is

right (left) closed in R if whenever X is right (left) essential in Y then X = Y.

Let 0 denote an arbitrary radical property, 0(R) the sum of all 0-ideals of

R, and p will symbolize a supernilpotent radical property (i.e, a hereditary

radical which contains the prime radical). Our terminology and definitions for

radicals will conform to that used in [GI and [u]. Many well known radicals

are supernilpotent: the prime (or Baer lower) radical P, the nil radical N,

the generalized nil radical N,, the Levitzki radical L, the Jacobson radical J,

and the Brown-McCoy radical G. We say R is essentially &radical if 0(R) is

R-essential in R. We will use E(0) to denote the class of essentially 0-radical

rings. In the terminology of [ U], E(0) is called the essential cover of the class

of &radical rings. If I( a R and I( E £(0), then we say I( is an essentially

9-radical ideal of R.

1. The Class E(9 )

As our initial motivation for studying the class E(0), let us observe that

an immediate consequence of Fisher's Theorem 2.3 in [8] is that the class of

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3

RINGS WHICH ARE ESSENTIALLY SUPERNILPOTENT 1065

essentially nilpotent rings is & ( P ) . One then may wonder "what happens

when P is replaced by other radicals?" One can immediately observe that

a subdirectly irreducible ring, prime ring, or a right uniform ring is either

0-semisimple or essentially 9-radical. The split-null extension S(R, M ) of the

(R, R)-bimodule M by R is in E ( P ) if M is either left or right faithful [a]. Any local ring which is not a division ring is in E( J ) , and any ring with unity

which has a unique maximal ideal [B] is in E(G). Many more examples will

be provided in the sequel.

From [11, Lemma 11, every 9-radical ring R has an essential extension

R' with unity element, such that each ideal of R is an ideal of R' too. In

general, R' is a homomorphic image of the cannonical embedding of a ring

into a ring with unity (i.e., the Dorroh extension). We begin this section by

determining conditions which guarantee that the Dorroh extension of a radical

ring yields a ring which is essentially radical. Using this result we construct

several examples of essentially 9-radical rings for 9 =P, N, L, and G . We

conclude this section with a theorem which shows that E(p) is closed relative

to ideals, extensions, direct sums, direct products, finite subdirect products,

and upper triangular matrix rings.

We use (R;) for the Dorroh extension of R; and /(n) denote the ring

of integers and integers modulo n, respectively.

Proposition 1.1 Let R be a 9-radical ring with char(R) = 0.

(i) If for each 0 # x E R and nonzero integer k there exists b E R such

that either xb+ kb # 0 or bx + kb # 0, then (R;) E E(0).

(ii) If 9() = 0 and (R;) E E(9), then for each 0 # x E R and nonzero

integer k there exists b E R such that either xb + kb # 0 or bx + kb # 0.

Proof. (i) Let 0 + (x, k) E (R;). We will show that < (x, k) >(R;z ) n e((R;

)) # 0. If k = 0, then (x, k) E (R, 0). If x = 0, there exists b E R such that

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3

1066 BIRKENMEIER

0 # (x, k)(b, 0) = (kb,O) E (R,O). If x # 0 and k # 0, then (by hypothesis)

there exists b E R such that either 0 # ( b ; ~ ) ( x , k) = (bx + kb,O) E (R,O), or

0 # (x, k)(b, 0) = (xb + kb, 0) E (R, 0). Since (R, 0) C 8((R;)), it follows that

(R;) E w .

(ii) Since 8() = 0, O((R;)) = (R,O) [El. Let (x, k) E (R;) such that

x # O and k # 0. There exists 0 # (y, 0) E< (x, k) > ( ~ ; q . Assume that

for all b E R, xb+ kb = 0 = bx + kb. Then (y,O) = ~(a; ,m;)(x ,k)(c ; ,n i ) =

(C m;n;)(x, k), a contradiction.

Example 1.2 Assume (R, +) is torsionfree.

(i) If R is nilpotent, then (R;) E E(P) .

(ii) If R is L-radical, then (R;) E E(L).

(iii) If R is nil, then (R;) E E(N) .

In all of the above cases, use Proposition 1.1 (i) with x of nilpotent index

n and take xn-l for b.

Example 1.3 [fi, pp. 109-1 11,120] Let W be the ring of endomorphisms

of an infinite dimensional vector space V over a field of characteristic zero.

Let R be the subring of all endomorphisms of finite rank. Then R is a simple

ring (without unity) such that J ( R ) = 0 but G(R) = R. By Proposition 1.1

(R;) is essentially G-radical. However from [E l , J ( (R ; ) ) = 0.

Lemma 1.4 The class E(8) has the following properties:

(i) Let I a R such that I is R-essential in R. If I E E(8), then R E E(8)

(i.e., E(8) is essentially closed).

(ii) &(8) has the inductive property: if B1 B2 C . C B, - . is a

chain of essentially 8-radical ideals of R, then U, B, E E(8).

(iii) If I( is an essentially 8-radical ideal of R, then B(K) is R-essential in

I(. Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3


Proof. Parts (i) and (ii) are immediate. Part (iii) follows from the fact that if

1 a R and I & K then I a A'.

Lemma 1.5 If PL 8, then the class E(8) has the following properties:

(i) If Ii' a R and 8(K) is R-essential in I(, then II' E E(8).

(ii) If {R, : a E A} E(8) then S E E(8), where S is either the direct

product or the direct sum of the R, ( a E A).

(iii) If X a I( a R and X E E(B), then (X)R E E(8).

(iv) If X is a right ideal of R with 8(X) C B((X)J and X E E(8), then

W R E w .

Proof. (i) Let 0 # H a K . Then < H >$C H . If < H >;= 0 then 0 # < H >R

8(K). If < H >;# 0, then < H >; n 8(K) # 0. Hence 0 # H n 8(K).

Thus K is essentially 8-radical.

(ii) Assume I a S such that 8(S) n I = 0, where S is the direct product

of R, (a E A). Let R, be the image of the canonical injection of R, into

S. Then 8 ( X ) n I = 0. H e n c e z n I = 0, for all a E A. Thus 12 = 0. So

I C O(S) n I = 0. Therefore, S E E(8). The proof for S a direct sum is similar.

(iii) Let 0 # I a K such that I 2 (X)R. If 13 = 0, then I n 8 ( ( X ) R ) # 0. So

assume 13 # O. T ~ U S o # 2 (x): 2 X. Hence o # ~ ~ n e ( x ) E In8((X)R).

Therefore 8((X) R) is I(-essential in (X) R. By part (i), (X) E &(8).

(iv) There exists B a R such that B C (X)R, B n (8(X)), = 0, and

B $ (O(X))R is R-essential in (X)R. Hence B n X = 0. Thus X B = 0, so

B2 = 0. Consequently, B $ (B(X))R C 8((X),). By part (i), (X)R E l (8) .

The following example shows that PL 8 is necessary in Lemma 1.5 (i).

Example 1.6 Let D ( R ) denote the divsible radical of R (i.e., D ( R ) is

the largest divisible additive subgroup of R), and the rational field. Let

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3

BIRKENMEIER

the structural matrix ring and I( = [: : 1. Then 0 0 0

I( a R and D ( K ) = 0 0 is R-essential in I<; but K 4 &(D) because [: : :I InD(I() = 0, where I = 0 0 0 is an ideal of I(. I: : 0 1

We note that Lemma 1.5 (ii) guarantees that is closed under direct

products. However, in general, supernilpotent radical classes are not closed

under direct products (e.g., the class of prime radical rings is not closed under

direct products [5]).

The following example shows that 8(X) 8((X)R) is necessary in Lemma

1.5 (iv).

F F Example 1.7 Let R = [ i. ] and X = [ f ] , where i. is a held.

Then X E E(P) , but (X)R = R @ E(P) .

Lemma 1.8 If X, Y, I( a R such that Ii' C X n Y and X/K is R / K

-essential in Y/K, then X is R-essential in Y.

Proof. The result is immediate if I( = 0. So assume I( # 0. Let 0 # B a R

such that B 5 Y. If B 2 K , then X n B # 0. So assume B I(. Then

X / K n ( B + K ) / K # 0. Hence there exists x E X , b E B, and k E I( such

that x $I< and b = x + k # 0. Hence X n B # 0 in all cases.

Lemma 1.9 If 0 is a hereditrary radical, then the class E(8) has the fol-

lowing properties:

(i) If I< a R such that K and R / I - are in E(B), then R E E(8) (i.e., E(0) is

closed under extensions).

(ii) Let S be either a direct sum or a direct product of rings R, (a E A).

If S E E(0) then R, E E(O), for all cu E A. Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3


Proof. (i) Let X a R such that X / K = 8 (R/I(). Hence 8 (R) C X. Let B a R

such that B C X and B n 8 (R) = 0. Since 8 ( K ) is R-essential in Ii' and

0 (I<) 2 8 (R), then B n K = 0. Now (B + I() /I( is an ideal of X/K. Since 8

is hereditary, ( B + I() /I- E B E &(8). Hence B = 0. So 8(R) is R-essential

in X. Since X/ I< is R/ I(-essential in R/K, Lemma 1.8 yields R E E(8).

(ii) Let 0 # B a R,. Take B and R, to be the images of B and R,,

respectively, under the cannonical injection of R, into S. Hence 0 # ~ n 8 ( S ) =

B(B) = B n O(R,). Thus 0 # B n O(R,). Consequently, R, E E(8).

The following lemma is an adaptation of Proposition 7.1 in [g, p. 1721.

Lemma 1.10 Let C be a hereditary class (i.e., I a R and R E C implies

I E C) which is closed under extensions. Then C is closed under finite subdirect

products.

Proof. Suppose I and I( are ideals of R such that I n I( = 0, and R / I and

R/K are in C. Then I E I/(I n I() E (I + K ) / K a R /K, so I E C. Since

R / I E C, R E C. The result follows by induction.

Theorem 1.11 &(p) has all the properties listed in Lemmas 1.4, and 1.5

plus the following properties:

(i) E(p) is a hereditary class.

(ii) E(p) is closed under extensions.

(iii) E(p) is closed under finite subdirect products.

(iv) Let S be either a direct sum or a direct product of rings R,(a E A).

Then S E E(p) if and only if R, E E(p), for all a E A.

(v) Let IC a R. Then K E &(p) if and only if p(K) is R-essential in K.

(vi) If R E E(p) then T E E(P), where T is the ring of n-by-n upper

triangular matrices over R.

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3

1070 BIRKENMEIER

Proof. (i) Let R E E(P) and Ii' a R. Assume 0 + I a R such that I & Ii'. Then

0 + I n p(R) & Ii' n p(R) = P(K). The result follows from Lemma 1.5 (i).

(ii) This part follows from Lemma 1.9 (i).

(iii) Parts (i) and (ii) and Lemma 1.10 yield this result.

(iv) This part is a consequence of Lemma 1.5 (ii) and Lemma 1.9 (ii).

(v) Lemma 1.4 (iii) and Lemma 1.5 (i) yield this result.

(vi) Let K be the ideal of T consisting of all strictly upper triangular

matrices. Then Ii' C p(T). Observe that T / K is isomorphic to a direct sum

of copies of R. The result follows from parts (ii) and (iv).

Note that there is no converse to part (vi) of Theorem 1.11. For let R be

a field, take n = 2, then T E E ( P ) but R 4 E(P) . Also, in general, E(p)

is not closed under homomorphic images. Observe that /(4) E E ( P ) , but

/(2) 4 W ) .

2. The R-essential closure of a supernilpotent radical

Clearly not every ring is in E(8), for a given 8. So one may naturally

ask, "is there a largest substructure of a ring which is in E(8), for a given

O?" From Lemma 1.4 (ii), we see that R contains an ideal which is maximal

among ideals in E(8). However for a supernilpotent radical more can be said.

We show that for every ring R and every p there exists a unique largest ideal

p(R) of R such that p(R) E E(p). This ideal generalizes the ideal E N ( R )

(i.e., the unique largest essentially nilpotent ideal of R) developed by Eslami

and Stewart in [ I] . In fact EN(R) is the same as F(R) (i.e. ,the special case

when p = P ) . Following the format of their paper we are able to extend their

results on essentially nilpotent rings and ideals to the more general context

of essentially p-radical rings and ideals. These results not only describe the

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3


properties of p(R), but also connect the behavior of p(R) to p(S), where R

and S are related rings. Throughout 6(R) denotes the sum of all essentially

&radical ideals of R. Henceforth Lemma 1.4 (iii) and Lemma 1.5 (i) will be

used implicitly where applicable.

Proposition 2.1 Let P I 0. Then:

(i) An arbitrary sum of essentially 0-radical ideals is an essentially 0-radical

ideal.

(ii) B(R) is the unique largest essentially 0-radical ideal of R.

(iii) If h : R + W is a ring homomorphism such that h(R) = W and

O(R) n ker h = 0, then ~(B(R)) C B(w). In particular, if h is an isomorphism,

then ~(B(R)) = B(w).

(iv) If X is a right (or left) ideal of R such that X n O(R) = 0, then K2 = 0

where K = ( X ) R n O(R). Thus if R is semiprime, then O(R) is right and left

essential in o(R).

Proof. (i) Let T = C Aj ( j E J ) be an arbitrary sum of essentially 0-radical

ideals of R. Let 0 # K a R such that K & T. If 0 # I( n Ak for some k E J ,

then 0 # I( n O(T). So assume 0 = K n A j , for all j E J . Let 0 # x E K .

Then x = C a j ( j E H), where H is a finite subset of J and aj E A j . Then

< x >;= 0. Again 0 # K n O(T). Hence T is an essentially 0-radical ideal of

R.

(ii) This part follows from part (i).

(iii) Let K = ker h. First we will show that O(R/K) is R/K-essential

in ((B(R) + I i ) / K ) +O(R/K). Let I / K a R/K such that I / K 2 ((B(R) + l i ) / K ) +O(R/K) and O(R/K) n I / K = 0. If ( I / K ) n ((B(R) + K ) / K ) = 0,

then ( I / K ) 2 = 0. Hence I / K & O(R/K). Thus I / K = 0. So assume

( I / K ) n ((B(R) + K ) / K ) # 0. Then I n B(R) 9 K. Hence I n O(R) K.

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3

1072 BIRKENMEIER

But (O(R) + K ) / K C O(R/K). Thus ( I / K ) n O(R/K) # 0, a contradiction.

Consequently, O(R/K) is R/K-essential in ((B(R) + K ) / K ) +O(R/K). Hence

(B(R) + K ) / K s $(R/K). Since R/K is isomorphic to W, then ~ ( B ( R ) ) -

O(W).

(iv) This part follows from a routine argument.

Proposition 2.2 (i) There exists an ideal S of R which is maximal among

p-semisimple ideals of R. Furthermore p(R) is the left and right annihilator of

any maximal p-semisimple ideal of R. Thus p(R) is both left and right closed

in R.

(ii) p(R) $ S is right and left essential in R, where S is any maximal

p-semisimple ideal of R.

Proof. (i) A routine Zorn's lemma argument yields an ideal S which is maximal

among p-semisimple ideals of R. Observe that p(R) r(S). Since S is a

semiprime ring, r(S) n S = 0. Let 0 # I( a R such that K C r (S) and

I( n p(R) = 0. By the maximality of S, (I< $ S) n p(R) # 0. Hence there exists

0 # k E I( and 0 # s E S such that 0 # k + s E p(R). So 0 # s E r (S) n S,

a contradiction. Thus r(S) is essentially p-radical. Therefore p(R) = r(S).

Similarly p(R) = I (S) .

(ii) Assume X is a right ideal of R such that X n (p(R) $ S) = 0. Then

X $ p(R) C l(S) = p(R). Hence X = 0, so p(R) $ S is right essential in R.

Similarly p(R) $ S is left essential in R.

Theorem 2.3 (i) p(R) = {x E R I if I a R such that ( x ) ~ n I # 0, then

I n P(R) # 0).

(ii) p(R) is a semiprime ideal of R and P ( R / ~ ( R ) ) = 0 = p(R/p(R)).

(iii) If X a R, then p(X) = X n p(R).

(iv) Let R and S be the direct product and direct sum of the rings R,

(a E A) , respectively. Then p(R) = n p(R,) and p(S) = $ p(R,). Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3


(v) If 0 # c = c2 E R and p(cRc) p(R), then < p(cRc) > R G p ( ~ ) and

p(cRc) 2 cp(R)c.

(vi) If R has unity, 0 # c = c2 E R, RcR = R, and p(cRc) cP(R)c, then

p(cRc) = cp(R)c.

Proof. (i) Let V denote {x E R I if I a R such that (x),n I # 0, then Inp(R) #

0). Clearly p(R) G V. So take x E V. Let 0 # BaR such that B C (x),. Thus

H # 0, where H = B n p(R). Hence 0 # H = B n (x), n p(R) = B n p(x),.

Thus (x) , E E(p). By Proposition 2.1, x E p(R). Consequently, p(R) = V.

(ii) Assume X/p(R) is an essentially p-radical ideal of R/p(R). By Lemma

1.9 (i), X is an essentially p-radical ideal of R. From Proposition 2.1, X/p (R)

= 0. Hence p(R/p(R)) = 0 = p(R/p(R)). Since all nilpotent ideals of R/p(R)

are contained in p(R/p(R)), p(R) is a semiprime ideal of R.

(iii) Let B be an ideal of X such that 0 # B C_ < p(X) >R. Thus

< B >LC B n < p(X) >; 2 B n p ( X ) . If < B >&= 0, then B p(X).

If 0 #< B >;, then B n p(X) # 0. In either case, p(X) is X-essential in

< p(X) >R. Hence p(X) =< p(X) >R. Thus p(X) is an essentially p-radical

ideal of R, so p(X) C X n p(R). Since p(R) is R-essential in p(R), then

p(X) = X n p(R) is R-essential in X n p(R). Thus, X f l p(R) is an essentially

p-radical ideal of R. Hence X n p(R) C p(X).

(iv) We will only consider the direct product case since the direct sum case

is proved similarly. By Lemma 1.5 (ii), n p(R,) C p(R). So let x E p(R),

p,(x) = x, (the cannonical projection), and Ii' a R,. Assume H # 0, where

H = (x , )~ . n K . Let H be the image of the cannonical injection of H into

R. First assume H n (x), # 0. Let Y = H n (x),. Then Y fl p(R) # 0. As is

well known, p(R) C n p(R,) [l4, p. 1151. Hence Y n p(R,) # 0. Therefore

0 # I( n p(R,). By part (i), x, E p(R,). So, in this case, p(R) C: np(R,).

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3

1074 BIRKENMEIER

Now assume H n ( x ) ~ = 0. Then H ( x ) ~ = 0. So H~ = 0. Hence H 2 = 0.

Thus H c p(R,). So 0 # Ii' n p(R,). Again by part (i), p(R) 2 n p(R,).

Therefore, for all cases, p(R) = fl p(R,).

(v) We will show that < p(cRc) >R is an essentially p-radical ideal of R.

Let I( be a nonzero ideal of R such that Ii' C< p(cRc) >R. Then cKc

c < p(cRc) >R c C p(cRc). If 0 # cIi'c, then 0 # CKC n p(cRc). Now

o # CKC c K n < ~ ( c R c ) >R n p(R) = Ii' n p < ~ ( c R c ) >R. If CKC = 0,

then K 3 = 0. Hence I( C p < p(cRc) >R. Thus, < p(cRc) >R is essentially

p-radical. Hence, p(cRc) c< p(cRc) >RC_ p(R). Therefore p(cRc) C cp(R)c.

(vi) From part (v), we need only show that cp(R)c p(cRc). We will

proceed by showing cp(R)c is an essentially p-radical ideal of cRc. Let H be

a nonzero ideal of cRc such that H c cp(R)c. Then < H >R C p(R). Let

Ii' =< H >R n p(R) # 0. Suppose cli'c = 0. Then RIi'R = RcRKRcR =

RcIi'cR = 0. But R has a unity. Hence cKc # 0. Thus 0 # cKc C H n

cp(R)c = H n p(cRc). Therefore cp(R)c C p(cRc).

Parts (i) and (ii) of Proposition 2.1 and parts (ii), (iii), (v), and (vi) of

Theorem 2.3 were derived by Eslami and Stewart in [I] for the case when

p =P. In unpublished work, the author had also discovered parts (i) and

(ii) of Proposition 2.1 and parts (ii) and (iii) of Theorem 2.3 for p = P , prior

to seeing the paper by Eslami and Stewart. The remaining results and their

proofs in this section are direct extensions of the results and proofs by Eslami

and Stewart in [ '71. To introduce the background for these results we quote

from their paper:

"If R and S are rings with the same identity and R C_ S, then S is a

free normalizing extension of R and S is a free right and left R-module with

a basis X such that X R = Rx for all x E X. Note that in this case each

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3


x E X determines an automorphism 0, of R defined by xO,(r) = rx for all

T E R. A free normalizing extension S of R satisfies the essential condition if

whenever U V are ideals of S with U S-essential in V and I a R such that

IV # 0, then I V n U # 0. If S is a free centralizing extension of R; that is,

8, is the identity automorphism for all x E X, then certainly S satisfies the

essential condition because in this case I V a S . Also, if G is a finite group

of automorphisms of R and R has no IGI-torsion, then the crossed product

R * G satisfies the essential condition. This is because a minor modification of

the proof of Lemma 1.2 (ii) in Passman [El shows that if U and V are ideals

of R * G with U R * G-essential in V, then U is essential as an R - R * G

subbimodule of V."

Theorem 2.4 Let S be a free normalizing extension of R which satisfies

Proof. The proof is virtually the same as that of Theorem 3 in [I], where our

"p" and "p" replace their N and E N , respectively.

Let M,(R) denote the full ring of n-by-n matrices over R.

Corollary 2.5 Let p be a supernilpotent radical such that p(Mn(W)) =

M,(p(W)), whenever W is a ring with unity. Then p(Mn(R)) = Mn(p(R)),

where R does not necessarily have a unity.

Proof. This proof is analogous to the proof of Corollary 4 in [TI, where we use

"p" , " ~ " , and Theorem 2.3 in place of their "N" , "EN", and Proposition 2,

respectively.

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3

1076 BIRKENMEIER

As is well known [el or [l9] and [2, p.511, Corollary 2.5 can be applied to

P, J, G, and L. When p =P then Corollary 2.5 coincides with Corollary 4

of [I].

Corollary 2.6 Let p be a supernilpotent radical such that p(M,(W)) =

Mn(p(W)) and p(cWc) = cp(W)c whenever W is a ring with unity and 0 #

c = c2 E W such that WcW = W. If R and S are rings with unity which are

Morita equivalent, then R E &(p) if and only if S E &(p).

Proof. This follows immediately from Theorem 2.3 (vi) and Corollary 2.5.

Proposition 2.7 Let c E R such that c = c2.

(i) If every one-sided ideal of a O-ring is a O-ring, then cO(R)c C O(cRc).

(ii) Let 9 be a hereditary radical such that for any ring A, O(A) contains

all one-sided 9-ideals of A. Then O(cRc) C_ O(R).

Proof. (i) Observe that cO(R) is a O-ring, and that cO(R) a cR. Hence cO(R) C_

O(cR). Now cO(R)c is a left ideal of cR contained in O(cR). So cO(R)c is a

O-ring and cO(R)c a cRc. Therefore cO(R)c O(cRc).

(ii) Since O(cRc) is a O-ring and a left ideal of cR, then O(cRc) C O(cR). Let

K be the right ideal of R generated by B(cRc). Then K C O(cR) and K a cR.

Thus K is a 8-radical ring. Hence I< C O(R). Consequently, O(cRc) C O(R).

It is well known that for c = c2 the equation p(cWc) = cp(W)c is true

for p =P and J . Using Proposition 2.7 and [2, pp.49-511, we see that it also

holds for L. Hence Corollary 2.6 can be applied to P, J, and L. For p =P,

Corollary 2.6 coincides with Corollary 7 of [I].

Let R[x] denote the ring of polynomials over R in the commuting indeter-

minate x.

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3


Corollary 2.8 (i) If p ( R ) [ x ] & P ( R [ x ] ) , then p ( R ) [ x ] G p ( R [ x ] ) .

(ii) If / @ [ X I ) S p(R)[xI , then p ( R [ x l ) C p(R)[x l .

Proof. The proof is analogous to the proof of Corollary 6 in [I] , where we

use "p" , "j7"' and Theorem 2.4 in place of their " N " , " E N " , and Theorem 3,

respectively.

Using results in [I] and Corollary 2.8, we have that p ( R [ x ] ) = p ( R ) [ x ] for

p =P or L and p ( R [ x ] ) C p ( R ) [ x ] for p =J . When p =P, Corollary 2.8

coincides with Corollary 6 of [I].

Corollary 2.9 Let G be a finite group of automorphisms of R such that

R has no IGI-torsion. If p(R * G ) = p ( R ) * G, then p(R * G ) = p(R) * G, where

R * G is the crossed product.

Proof. This proof is analogous to the proof of Corollary 5 in ['7] using the

aforementioned replacements.

Corollary 2.10 Let R be a ring with unity, G a finite group of auto-

morphisms of R such that IGI is invertible in R, and c = IGI-'C,,~ g . If

p(cRGc) & p ( R G ) and p(RG) = p ( R ) G , then F ( R ~ ) p(R), where R G is the

skew group ring.

Proof. Observe that c is an idempotent in R G and from [ l5, Lemma 6.11

c ( R G ) c = RGc N RG. By Theorem 2.3 (v), p(RGc) p ( R G ) and p ( R G ) =

p ( R ) G by Corollary 2.9. Since p(RGc) = p ( R G ) c it follows that p(RG) & p ( R ) .

Using [l5, Theorem 7.11, if 1 ~ I - l E R, then Corollaries 2.9 and 2.10 hold

for p = J . When p = P, then Corollaries 2.9 and 2.10 coincide with Corollaries

5 and 8 of [ I ] , respectively.

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3

3. The socle and p(R)

BIRKENMEIER

In this section we will consider the behavior of p(R) relative to minimal

right ideals. We denote the sum of all minimal right ideals of R by Soc(R),

and call it the socle of R. For a minimal right ideal X of R, the homoge-

neous component is symbolized by C(X) [ l2, pp.63-661. Let ZT (R) denote the

right singular ideal of R [lo, pp.30-311 and &(R) the ideal of R defined by

Z2(R)/ZT(R) = ZT(R/ZT(R)) [lo, ~ . 37 ] . We note that Z2(R) is right closed in

R and a right essential extension of ZT(R).

Lemma 3.1 Let X be a minimal right ideal R and S a maximal p-

semisimple ideal of R.

(i) The following conditions are equivalent:

( 4 x c m; (b) X n S = 0;

(c) either X C_ p(R) or X is R-isomorphic to a nilpotent right ideal of

R.

(ii) If X n p(R) = 0, then the following conditions are equivalent:

(a) X C S;

(b) < X >R is a minimal ideal of R.

Proof. (i) By Proposition 2.2, the equivalence of (a) and (b) is immediate.

Now assume X p(R) and X p(R). Then X n p(R) = 0. Hence there

exists e = e2 such that X = eR. First suppose that p(R)eR # 0. Then there

exists y E p(R) such that yeR # 0. Hence left multiplication by y provides

an R-isomorphism from eR to the nilpotent right ideal yeR. Now consider

p(R)eR = 0. Let K =< eR >R n p(R). Since X C_ p(R), K f 0 but K2 = 0.

Note that < eR >R= C(eR), hence eR is again R-isomorphic to a nilpotent

right ideal. Thus (a) implies (c). Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3


Now suppose either X 5 p(R) or X is R-isomorphic to a nilpotent right

ideal of R. If X p(R), then (c) implies (a). So assume X n p(R) = 0 and X

is R-isomorphic to a nilpotent right ideal of R. Then there exists e = e2 E R,

such that X = eR. Since < eR >R = C(X), then < eR >R n p(R) # 0. Let

O # I a R such that I c< eR >R. If I n P(R) = 0, then there exists c = c2 E I

such that < cR >R = < eR >R, a contradiction. Hence < eR >R is essentially

p-radical, so eR & p(R). Therefore (c) implies (a).

(ii) This part is proved using arguments similar to those used in part (i).

Theorem 3.2 Let X be a minimal right ideal of R and S a maximal

p-semisimple ideal of R. Then:

(i) Soc(R) = A $ B, where A = p(R) n Soc(R) and B = S n Soc(R);

(ii) if X & A, then C(X) & A and either X C p(R) or X is R-isomorphic

to a nilpotent right ideal of R;

(iii) if X B, then < X > R = C(X) is a minimal ideal of R and X = eR,

where e = e2 E R;

(iv) A = I( $ D (right ideal direct sum), where K = p(R) n Soc(R) and

D contains no nonzero ideals of R.

Proof. For part (i), observe that if X S then X n S = 0 = XS. Hence

X C p(R) = r(S). Thus Soc(R) = A $ B. For part (ii), assume X & A. If

X = eR, where e = e2 E R, then < X >R = C(X) C A. Otherwise X is

nilpotent. Let Y be a right ideal of R such that Y is R-isomorphic to X. By

Lemma 3.1 (i), Y C A. Thus C(X) C_ A in all cases. The remainder of the

proof is immediate from Lemma 3.1.

Proposition 3.3 Let R be a ring such that Soc(R) is right essential in R.

Then Z2(R) C P(R) = 5 (R) C F(R) for any supernilpotent radical p.

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3

1080 BIRKENMEIER

Proof. Let 0 # x E ZT(R). Then < x > R contains a minimal right ideal

Y. Since ZT (R) contains no nontrivial idempotent element of R, then Y2 = 0.

Hence ZT(R) is essentially P-radical. Since Z2(R) is a right essential extension

of ZT (R), then Z2(R) &P(R) . A similar argument shows p ( R ) = J ( R ) . Now - J ( R ) & p(R) because PI p for any supernilpotent radical p.

4. Open problems

From [s], a class of rings is a semisimple class if and only if it is closed

relative to ideals, extensions, and subdirect products. By Theorem 1.11, &(p)

is closed relative to ideals, extensions, finite subdirect products, direct sums,

and direct prodtucts. Thus one of the referees posed the following problems:

(1) Is &(p) a semisimple class (probably not)?

(2) If not, a necessary and sufficient condition on p would be interesting to

assure that & ( p ) is a semisimple class of some radical 8.

Another question one may ask is: what conditions can be put on a universal

class of rings U so that &(8) is a semisimple class? (e.g., ZA is a finite class and

8 is a supernilpotent radical).

Finally we ask, "how does the theory of essentially p-radical rings and ideals

relate to the Koethe Conjecture?" For example, is the Koethe Conjecture

equivalent to J(R[x]) =N(R)[x] for all rings R? (see [n, pp.209-2111.

ACKNOWLEDGEMENT

The author is grateful for the comments and suggestions made by the

referees which lead to the present improved version of this paper. In particular,

open questions 1 and 2 were posed by one referee while Example 1.6 is due to

the other referee.

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3


REFERENCES

[I] S. A. Amitsur, Radicals of polynomial rings, Canad. J . Math. 8 (1956),

355-361.

121 , Nil radicals. Historical notes and some new

results, Coll. Math. Soc. Janos Bolyai 6, Rings, Modules and Radicals,

Keszthely (Hungary) ,(1973), 47-65.

[3] G. F. Birkenmeier, Split-null extensions of strongly right bounded rings,

Publicacions Matematiques 33 (1989), 37-44.

[4] G. F. Birkenmeier, When does a supernilpotent radical essentially split-

off?, submitted.

[5] G. F. Birkenmeier, H. Heatherly, and E. Lee, Prime ideals and prime

radicals in near-rings, Monatshefte fiir Mathematik, to appear.

[6] N. J. Divinsky, Rings and Radicals, University of Toronto Press, Toronto,

1965.

[7] E. Eslami and P. Stewart, Two-sided essential nilpotence, Internat. J .

Math. & Math. Sci. 15 (1992), 351-354.

[8] J . W. Fisher, On the nilpotency of nil subrings, Canad. J . Math. 22

(1970), 1211-1216.

[9] B. J. Gardner, Radical Theory, Longman Scientific & Technical, Harlow,

1989.

[lo] K. R. Goodearl, Ring Theory Nonsingular Rings and Modules, Marcel

Dekker, New York, 1976.

[ll] G. A. P. Heyman and C. Roos, Essential extensions in radical theory for

rings, J. Austral. Math. Soc. 23 (Series A) (1977), 340-347. Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3

1082 BIRKENMEIER

[12] N. Jacobson, Structure of Rings, Colloquium Publication vol. 37 Amer.

Math. Soc., Providence, 1964.

[13] L. C. A. van Leeuwen, C. Roos, and R. Wiegant, Characterizations of

semisimple classes, J . Austral. Math. Soc. 23 (Series A) (1977), 172-182.

[14] J . D. P. Meldrum, Near-Rings and their Links with Groups, Pitman,

Marshfield, MA, 1985.

[15] D. S. Passman, It's essentially Maschke's theorem, Rocky Mountain J .

Math. 13 (1983), 37-54.

[16] B. D. L. Rosa and G. A. P. Heyman, A note on radicals and the Dorroh

extension, Arch. Math. 42 (1984), 516.

[17] L. H. Rowen, Ring Theory I, Academic, Boston, 1988.

[18] M. Satyanarayana and M. G. Deshpande, Rings with unique maximal

ideals, Math. Nachr. 87 (1979) 213-219.

[19] F. A. Szisz, Radicals of Rings, Wiley, New York, 1981.

Received: February 1993

Revised: May 1993 and September 1993

Dow

nloa

ded

by [

"Uni

vers

ity a

t Buf

falo

Lib

rari

es"]

at 0

0:11

02

May

201

3

rings which are essentially supernilpotent

Documents