This article was downloaded by: [UQ Library]On: 20 November 2014, At: 20:45Publisher: 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

Certain classes of ideals in group rings IIN.J. Groenewald a & G.A.P. Heyman ba University of Port Elizabeth , Republic of South Africab University of Bloemfontein , Republic of South AfricaPublished online: 27 Jun 2007.

To cite this article: N.J. Groenewald & G.A.P. Heyman (1981) Certain classes of ideals in group rings II, Communicationsin Algebra, 9:2, 137-148, DOI: 10.1080/00927878108822569

To link to this article: http://dx.doi.org/10.1080/00927878108822569

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purposeof the Content. Any opinions and views expressed in this publication are the opinions and views of theauthors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should notbe relied upon and should be independently verified with primary sources of information. Taylor and Francisshall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, andother liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relationto or arising out of the use of the Content.

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 anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

COMMUNICATIONS I N ALGEBRA, 9 ( 2 ) , 137-148 (1981)

CERTAIN CLASSES OF IDEALS I N GROUP RINGS I1

N. J . Groenewald U n i v e r s i t y of P o r t E l i z a b e t h , Republic of South A f r i c a

G.A.P. Heyman U n i v e r s i t y of Bloemfontein, Republic of South A f r i c a

I n [5] we determined r e l a t i o n s h i p s between c l a s s e s of i d e a l s i n

RH(R) and t h e group r i n g RG. These r e s u l t s were t h e n a p p l i e d t o

o b t a i n r e l a t i o n s between d i f f e r e n t r a d i c a l s of t h e r i n g R and t h e

group r i n g RG (semigroup r i n g RS). I n t h i s n o t e i t i s o u r pur-

pose t o c o n s t r u c t t h e upper r a d i c a l c l a s s determined by t h e c l a s s

of a l l s t r o n g l y prime r i n g s and t o determine some r e l a t i o n s

between t h i s r a d i c a l of t h e r i n g R and t h e semigroup r i n g RS.

F i n a l l y , we determine r e l a t i o n s between t h e g e n e r a l i z e d n i l

r a d i c a l h' of t h e r i n g R and t h e semigroup r i n g RS. E

5 1 . THE STRONGLY PRIME RADICAL

I n t h e i r paper [6] Handelman and Lawrence i n t r o d u c e d s t r o n g l y

prime r i n g s , a n o t i o n which t u r n s o u t t o be t h e same a s t h a t of

a b s o l u t e l y t o r s i o n f r e e (ATF) r i n g of Rubin [ 1 4 ] .

Dow

nloa

ded

by [

UQ

Lib

rary

] at

20:

45 2

0 N

ovem

ber

2014

138 GROENEWALD AND HEYMAN

I n t h i s s e c t i o n i t i s o u r purpose t o c o n s t r u c t t h e upper r a d i c a l

c l a s s determined by t h e c l a s s of a l l s t r o n g l y prime r i n g s . Our

main r e s u l t i n t h i s s e c t i o n i s t h a t t h i s r a d i c a l i s s p e c i a l i n

t h e s e n s e of Andrunakievic [ I ] . Other p r o p e r t i e s shared by some

of t h e most prominent r a d i c a l s a r e then proved.

We work e n t i r e l y i n t h e ca tegory of a s s o c i a t i v e r i n g s and we pre-

suppose a f a m i l a r i t y w i t h t h e te rmin i logy and b a s i c r e s u l t s of

r a d i c a l t h e o r y , most o f which can be found i n [ 3 ] .

We say a r i n g R i s r i g h t s t r o n g l y prime i f g i v e n any nonzero a E R,

t h e r e e x i s t s a f i n i t e s e t { x ,... , x ) i n R such t h a t { a x , . . . a x 1 1 n

has ze ro r i g h t a n n i h i l a t o r . L e f t s t r o n g l y prime r i n g s a r e d e f i n e d

analogously and a r i n g i s s a i d t o be s t r o n g l y prime i f i t i s bo th

l e f t and r i g h t s t r o n g l y prime. However, we w i l l g e n e r a l l y work

on t h e r i g h t , and ' s t r o n g l y prime' w i l l denote r i g h t s t r o n g l y

prime. An a l t e r n a t e d e f i n i t i o n f o r a ( r i g h t ) s t r o n g l y prime r i n g

is one i n which every twosided i d e a l c o n t a i n s a f i n i t e s e t whose

r i g h t a n n i h i l a t o r i s ze ro ( c f . [ 6 ] ) .

Let us now d e f i n e a s fo l lows .

D e f i n i t i o n 1 . Let R be a r i n g and Q an i d e a l of R. Then Q i s

c a l l e d a s t r o n g l y prime i d e a l of R i f f o r every x E C (Q) t h e r e R

e x i s t s a f i n i t e s u b s e t I of R such t h a t i f r E R and x I r 5 Q t h e n

r E Q.

D e f i n i t i o n 2 . A s e t S of e lements of a r i n g R i s s a i d t o be a n s-

system i f f o r every x E S t h e r e e x i s t s a f i n i t e s u b s e t I of R such

Dow

nloa

ded

by [

UQ

Lib

rary

] at

20:

45 2

0 N

ovem

ber

2014

CERTAIN CLASSES OF IDEALS. I1

t h a t f o r every r E S we have xIr fl S f 4 . 4 is a l s o d e f i n e d t o

be an s-system. Without l o s s of g e n e r a l i t y , we can a l s o c a l l I

t h e . i n s u l a t o r of a .

It i s then c l e a r from t h e d e f i n i t i o n of a s t r o n g l y prime r i n g and

a s t r o n g l y prime i d e a l t h a t R/Q i s a s t r o n g l y prime r i n g i f and

on ly i f Q i s a s t r o n g l y prime i d e a l and t h a t Q is a s t r o n g l y prime

i d e a l of R i f and o n l y i f CR(Q) is an s-system. Furthermore, i t

i s easy t o show t h a t every s t r o n g l y prime i d e a l i s a prime i d e a l

and consequently every ( r i g h t ) s t r o n g l y prime r i n g i s a prime

r i n g .

We can now prove.

Lemma I . 1 . Let A b e a n i d e a l and P a s t r o n g l y prime i d e a l of t h e

r i n g R. Then P fI A i s a s t r o n g l y prime i d e a l of A.

Proof . Let b E CA(AfIP) = CR(P) fl A. S ince P i s a s t r o n g l y - prime i d e a l i n R and b E C (P) , t h e r e e x i s t s a f i n i t e s u b s e t

R

{xl ,. .. ,x 1 of R such t h a t i f c E CR(P) then b x . c E CR(P) f o r n 1

some 1 < j 5 n . Let d E CR(P) n A 2 CR(P), t h e n from t h e above

t h e r e e x i s t s an % 6 {xl , . . . , x 1 such t h a t bxkd 6 CR(P). For n

t h i s element of CR(P) t h e r e e x i s t s a f i n i t e s u b s e t { z , . . . , z 1 of 1 m

R such t h a t {b\dz t , . . . ,bxkdzmt} CA(P fI A) 4 f o r every 1

t E CK(P) n A. Now {xkdz >,... ,xkdzm) 2 A, s o we have proved

t h a t f o r every b E C (P fI A) t h e r e e x i s t s a f i n i t e s u b s e t A

{xkdzl ,. . . ,\dzm} i n A such t h a t {b\dz t , . . . ,b\dzmt) fl cA(P n A)

# 4 . Thus P I? A is a s t r o n g l y prime i d e a l i n A.

Dow

nloa

ded

by [

UQ

Lib

rary

] at

20:

45 2

0 N

ovem

ber

2014

140 GROENEWALD AND HEYMAN

Andrunakievic [ I ] c a l l e d a h e r e d i t a r y c l a s s If of prime r i n g s

s p e c i a l i f M s a t i s f i e s

(a) B E M , B an i d e a l of R and Ann(B) = { r € R I B ~ = ~ B = o } = O

imply R E 11.

Furthermore, h e c a l l e d a r a d i c a l special i f i t i s t h e upper r a d i -

c a l determined by a s p e c i a l c l a s s of r i n g s .

It was shown i n [9 ] t h a t (a) may be r e p l a c e d by

( a ' ) Every r i n g having an e s s e n t i a l i d e a l i n M belongs i t s e l f

t o M,

Theorem 1.1. The c l a s s M of a l l s t r o n g l y prime r i n g s i s a s p e c i a l

c l a s s of r i n g s .

Proof . We know t h a t a l l r i n g s i n M a r e prime (cf [ 6 ] ) . Let - then R € M and I any nonzero i d e a l of R. S ince R i s s t r o n g l y

prime, i t fo l lows t h a t (0) i s a s t r o n g l y prime i d e a l of R so t h a t

(0) n I = (0) i s a s t r o n g l y prime i d e a l of I accord ing t o Lemma

1 .1 . Hence I I s a s t r o n g l y prime r i n g so t h a t M i s a h e r e d i t a r y

c l a s s .

Let f i n a l l y R be a r i n g which c o n t a i n s a member I of M a s a n es-

s e n t i a l i d e a l . From [ 6 ] , P r o p o s i t i o n 1 V . I we have R € M. Hence

M is a s p e c i a l c l a s s indeed.

We know now t h a t UM, t h e upper r a d i c a l determined by M , is a

s p e c i a l r a d i c a l c l a s s so t h a t according t o [ l ] we can s t a t e .

C o r o l l a r y 1.1. I n any r i n g R, UM(R) = { ~ I ( I ~ R and R / I E M}, i . e .

UM(R) = Cn111 i s a s t r o n g l y prime i d e a l of R}.

Dow

nloa

ded

by [

UQ

Lib

rary

] at

20:

45 2

0 N

ovem

ber

2014

CERTAIN CLASSES OF IDEALS. I1 14 1

Regarding t h e p o s i t i o n of UM among t h e w e l l known r a d i c a l c l a s s e s ,

we can show.

Theorem 1 . 2 . fi UM c a$; UM c N . N UM and f o r r i n g s s a t i s - g'

f y i n g t h e ascending cha in c o n d i t i o n on l e f t i d e a l s B = UM = N .

Proof. Every s t r o n g l y prime r i n g i s prime, so we d i r e c t l y g e t - B 5 h l . It i s known t h a t every s imple prime r i n g i s s t r o n g l y

prime. So every s u b d i r e c t l y i r r e d u c i b l e r i n g w i t h an idempotent

h e a r t i s s t r o n g l y prime f o r M i s a s p e c i a l c l a s s and R i s an es-

s e n t i a l e x t e n s i o n of a s t r o n g l y prime r i n g . Hence U M ~ fi 4)- We know t h a t i n t h e connnutative c a s e (3 = UM and s i n c e fi # fib, we

have UM # fig. Obviously every prime r i n g wi thout d i v i s o r s of

ze ro i s s t r o n g l y prime, so t h a t UM c N . We know about a n i l r i n g g

which i s s t r o n g l y prime ( c f . [ 6 ] ) , so N - $ UM. From [7] we know

t h a t a r i n g s a t i s f y i n g t h e ascending cha in c o n d i t i o n on l e f t

i d e a l s has t h e descending cha in c o n d i t i o n on r i g h t a n n i h i l a t o r s

and from [ 6 ] we have t h a t every prime r i n g w i t h D.C.C. on r i g h t

a n n i h i l a t o r s i s a s t r o n g l y prime r i n g . Hence fi = UM = N .

It i s w e l l known t h a t t h e equa t ion R(Rn) = [R(R)], i s n o t s a t i s -

f i e d by every r a d i c a l c l a s s R . It was, however, proved i n [ I ]

t h a t any s p e c i a l r a d i c a l c l a s s determined by a s p e c i a l c l a s s M

f o r which R € M i f and on ly i f Rn E M whenever R has an i d e n t i t y ,

s a t i s f i e s t h e s t a t e d equa t ion . For s t r o n g l y prime r i n g s we prove.

Lemma 1.2. R is a s t r o n g l y prime r i n g i f and o n l y i f R is s t rong- n

l y prime. Dow

nloa

ded

by [

UQ

Lib

rary

] at

20:

45 2

0 N

ovem

ber

2014

142

Proof . See [ 6 ] , P r o p o s i t i o n 11.1 . -

GROENEWALD AND HEYMAN

P r o p o s i t i o n 1 . 1 . For every r i n g R, [UM(R)] = UM(Rn). n

Proof . I f R has an i d e n t i t y t h e r e s u l t fo l lows from [ I ] . I f R - does n o t have an i d e n t i t y , i t can be embedded i n a r i n g A wi th

i d e n t i t y . R i s an i d e a l i n t h e r i n g A. Since UM i s h e r e d i t a r y ,

UM(R) = R n uM(A). By t h e c a s e a l r e a d y proved, we have

UM(An) = [ U M ( A ) ] ~ . Hence, s i n c e R i s an i d e a l i n A we have n n'

UM(R,) = Rn "M(An) = Rn "UM(A)In = [R fl UM(A)], = [UM(R)],.

52. THE STRONGLY PRIME RADICAL I N GROUP RINGS

Let R b e a r i n g w i t h i d e n t i t y and S a semigroup. The semigroup

r i n g of S o v e r R w i l l be denoted by RS. I n t h i s s e c t i o n we ob-

t a i n c e r t a i n r e l a t i o n s between t h e s t r o n g l y prime r a d i c a l of t h e

r i n g R and t h e semigroup r i n g RS. Furthermore, i f H i s a c e n t r a l

subgroup of t h e group G such t h a t G/H i s a r igh t -ordered group

then we show t h a t UM(RH) *RG = UM(RG) where R i s any r i n g w i t h

i d e n t i t y .

P r o p o s i t i o n 2.1. Le t R b e a s t r o n g l y prime r i n g w i t h i d e n t i t y .

I f S i s an u.p. semigroup w i t h i d e n t i t y element , then RS i s a

s t r o n g l y prime r i n g .

Proof . The proof uses t h e same technique a s t h e proof of Theorem - 2 .1 of 1 4 1 .

Let R be a r i n g and l e t ~ [ x ] be t h e r i n g of a l l polynomials i n a

c o m u t a t i v e inde te rmina te x over R.

Dow

nloa

ded

by [

UQ

Lib

rary

] at

20:

45 2

0 N

ovem

ber

2014

CERTAIN CLASSES OF IDEALS. I1

C o r o l l a r y 2.1 ( c f . [14] ) . I f R i s a s t r o n g l y prime r i n g w i t h

i d e n t i t y , t h e n ~ [ x ] i s a s t r o n g l y prime r i n g .

Proof . This i s c l e a r f rom P r o p o s i t i o n 2.1. - P r o p o s i t i o n 2.2. L e t R b e a r i n g w i t h i d e n t i t y and S a semigroup

w i t h i d e n t i t y element e . I f A i s a s t r o n g l y prime i d e a l i n RS,

t h e n A n R i s a s t r o n g l y prime i d e a l i n R.

Proof . See [4] , P r o p o s i t i o n 3 . 1 . - Theorem 2.1. I f R i s a r i n g w i t h i d e n t i t y and S a n u . p . semigroup

w i t h i d e n t i t y element , t h e n UM(R)S = UM(RS).

P roof . Le t Q b e a s t r o n g l y prime i d e a l i n R. From t h e isomor- - phism R/QS= RS/QS and P r o p o s i t i o n 2.1 we have QS a s t r o n g l y prime

i d e a l i n RS. The proof now f o l l o w s from P r o p o s i t i o n 2 . 1 and [ 5 ] ,

Theorem 2.1.

Coro l la ry 2.2. I f R is a r i n g w i t h i d e n t i t y , then

UM (R) [x] = UM (R [x] ) .

Let G be any group and R a r i n g w i t h i d e n t i t y . For each x t G

- 1 a n d a € R G w e d e f i n e X : R G + R G b y a X = x ax. From 1131 we

have t h a t con juga t ion i s always a n automorphism of RG. Now, i f

K is a s u b s e t of RG, then K is s a i d t o b e G-invariant i f K~ = K

f o r a l l x € G. I n p a r t i c u l a r , i f H a G , t h e n RG i s G-invariant

and i f I i s an i d e a l of RG, then I i s G-invariant .

P r o p o s i t i o n 2 . 3 . Let R b e a r i n g w i t h i d e n t i t y and H a c e n t r a l

Dow

nloa

ded

by [

UQ

Lib

rary

] at

20:

45 2

0 N

ovem

ber

2014

144 GROENEWALD AND HEYMAN

subgroup of t h e group G. I f A i s a s t r o n g l y prime i d e a l i n RG,

then RH n A i s a s t r o n g l y prime i d e a l i n RH.

Proof . The proof u s e s t h e same technique a s t h e proof of Propo- - s i t i o n 3.1 of [ 4 ] .

P r o p o s i t i o n 2 .4 . Le t R be a r i n g w i t h i d e n t i t y and l e t H Q G such

t h a t G / H i s a r i g h t o rdered group. I f Q i s a G-invariant s t rong-

l y prime i d e a l i n RH, t h e n Q'RG i s a s t r o n g l y prime i d e a l i n RG.

Proof . From [13] Q'RG i s an i d e a l i n RG. Let T be a t r a n s v e r s a l - of H i n G and suppose Q i s a s t r o n g l y prime i d e a l i n RH. Let

n m .. .~-

a E RG - Q-RG. We can w r i t e un ique ly a = 1 a . g . + ,I Bjhj where i= I J = I

g l , g 2 , . . . Y gn and h , h 2 , ..., h a r e elements of T and where m

a. C RH - Q and B € Q , 1 5 i 5 n and 1 5 j 5 m. L e t f u r t h e r L j

g l < g2 < ... < gn. S i n c e Q i s a s t r o n g l y prime i d e a l i n RH, a n

h a s an i n s u l a t o r I = {y , y 2 , . . . , y s ) i n RH. We show t h a t t h e 1

g n g n f i n i t e s e t J = { (Y ) , . . . , (yS) i s an i n s u l a t o r f o r a. Suppose

t h e r e e x i s t s an element b E RG - Q*RG such t h a t aJb 5 Q-RG. S ince

b f RG - Q*RG we can w r i t e uniquely b = nivi + A . w . where i= 1 j = l J J

v < v < ... < v and w l , ..., w a r e elements of T and where 1 2 P 9

t RH - Q and h E Q , 1 5 i 5 p and 1 5 j 5 q. L e t g , v t be t h e i j

maximal element of t h e s e t f g v . , g v 1. Then g . v < gnvt f o r n 1 ' n p 1 j

i , n , ; I < i $ n and 1 5 j 5 p. S ince Q'RG i s an i d e a l n

and aJb 5 QwRG, we have ( I a ig i ) J ( f r i v i ) 5 QWRG. Furthermore i= 1 i=i - 1 - 1

g n g n g a g (yi) T ~ V ~ = ~1 Y . ( r t ) gnvt = anyi ( r t ) nhg where h € H and

n n n 1 Dow

nloa

ded

by [

UQ

Lib

rary

] at

20:

45 2

0 N

ovem

ber

2014

CERTAIN CLASSES OF IDEALS. I1 145

From t h e above and t h e uniqueness of t h e elements of Q-RG we have - 1

g n anI(.rrt) h 5 Q. Since .rr (1 Q and s i n c e Q i s G-invariant ,

t

i - 1

gn ( r t ) h j! Q , t h i s c o n t r a d i c t s t h e choice of I. Hence J i s an

i n s u l a t o r f o r a and consequently Q-RG i s a s t r o n g l y prime i d e a l . I

Theorem 2.2. Let R be any r i n g w i th i d e n t i t y and H a c e n t r a l sub-

group of G such t h a t G / H i s a r i g h t ordered group. Then

UM(RG) = UM(RH) *RG.

Proof. This fo l lows from P ropos i t i ons 2.3 and 2.4 and a d i r e c t - a p p l i c a t i o n of Theorem 2 .3 of [ 5 ] .

Theorem 2.3. I f R i s a r i n g such t h a t R = R/UM(R) i s a conmutative

semisimple r i n g and G i s a so lvab l e group wi th no n o n t r i v i a l f i n i t e

normal subgroups, then UM(R)G = UM(RG) .

Proof. From [4 ] , P ropos i t i on 4.5 we have UM(EG) = (0). Further- - more XG Z RG/UM(R)G. Hence UM(RG/UM(R)G) = (0) and consequently

UM(RG) 5 UM(R)G. To show t h a t UM(R)G 5 UM(RG) , we have

UM(R)G = (I7 {Q : Q a s t r ong ly prime i d e a l i n R})G

As i n 141, P ropos i t i on 3.4 i t fo l lows from P ropos i t i on 2.2 t h a t

UM(R)G 5 fl { (PflR)G : P s t r o n g l y prime i d e a l i n RG)} f UM(RG) . Hence UM(R)G = UM(RG).

93. THE GENERALIZED NIL RADICAL I N GROUP RINGS

I n [ l l ] McCoy de f i ned an i d e a l P i n R t o be a completely prime

i d e a l i f ab € P impl ies t h a t a € P o r b € P, a and b be ing elements

of R. Thus, t h e i d e a l P i s completely prime i f and only i f C(P)

Dow

nloa

ded

by [

UQ

Lib

rary

] at

20:

45 2

0 N

ovem

ber

2014

146 GROENEWALD AND HEYMAN

i s a m u l t i p l i c a t i v e system ( s ee a l s o [2 ] ) . Let N denote t he g

gene ra l i z ed n i l r a d i c a l , i . e . t h e upper r a d i c a l determined by t h e

c l a s s of a l l nonzero r i n g s wi thout zero d i v i s o r s . From [ 2 ] we

know t h a t N (R) co inc ides w i th t h e i n t e r s e c t i o n of a l l t h e com- g

p l e t e l y prime i d e a l s of t h e r i n g R. I n t h i s s e c t i o n we determine

some r e l a t i o n s between t h e r a d i c a l N of t he r i n g R(RH) and t h e g

semigroup r i n g RS (group r i n g RG) .

Lemma 3.1. Le t R b e any r i n g and S a semigroup w i th i d e n t i t y e le -

ment. I f A i s a completely prime i d e a l i n RS, then A fl R i s a

completely prime i d e a l i n R.

P roof . Let a , b € R such t h a t ab € A fl R. S ince A i s a com- - p l e t e l y prime i d e a l i n RS and ab € A , c l e a r l y a € A n R o r

b € A fl R. Hence A fl R i s a completely prime i d e a l i n R.

Theorem 3.1. I f R i s any r i n g and S an u.p. semigroup w i th u n i t y ,

then N (R) S = N (RS) . g g

Proof. Let Q be a completely prime i d e a l i n R. From [ 101, - Theorem 2.1 (3) and t h e isomorphism R/Q S RS/QS, we have t h a t

QS i s a completely prime i d e a l i n RS. The r e s t now fo l lows

from [ 5 ] , Theorem 2.1.

P ropos i t i on 3.1. Let R be a r i n g w i th i d e n t i t y and H a G. I f Q

i s a completely prime i d e a l i n RG, then Q fl RH i s a G-invariant

completely prime i d e a l i n RH.

Proof. Adopt t he proof of Lema 3.1. - Dow

nloa

ded

by [

UQ

Lib

rary

] at

20:

45 2

0 N

ovem

ber

2014

CERTAIN CLASSES OF IDEALS. I1 147

P ropos i t i on 3.2. Le t R be a r i n g w i th i d e n t i t y and H a G such

t h a t G/H i s an u.p. group. I f Q i s a completely prime i d e a l i n

RH which i s a l s o G-invariant , then Q-RG i s a completely prime

i d e a l i n RG.

Proof. Let Q be a completely prime i d e a l i n RH and suppose Q*RG - i s no t completely prime. Hence t h e r e e x i s t s a , b C RG - Q 0 R G

such t h a t ab € Q-RG. Let T be a t r a n s v e r s a l of H i n G. Since n m

a , b € RG - Q*RG we can w r i t e uniquely a = I a . g . + 1 Rjhj and 1 1

i= 1 j = i

b = f alg; + @,!hi where g l , g , . .. , gn; h i , ..., h and m i= 1 j=i

g:, .. ., g i ; h i , .. ., h ' a r e elements of T and n . , a! C RH - Q , 4 1 1

B j , 63 € Q. Now by an argument s i m i l a r t o t h a t used i n t he

proof of [ 5 ] , P ropos i t i on 3 .2(1) we g e t :

-1

gi a i [ ( a i ) h] € Q, i € {1,2 ,..., n) , k € {1,2 ,..., and h C H.

- 1 Since Q i s G-invariant (a;)gi h f Q. This c o n t r a d i c t s the f a c t

t h a t Q i s a completely prime i d e a l i n RH and consequently Q0RG i s

a completely prime i d e a l .

Theorem 3 . 2 . I f R i s a r i n g w i th i d e n t i t y and H a G such t h a t

G / H i s an u.p. group, then N (RH) *RG = N g ( ~ ~ ) . g

Proof. This fo l lows from P ropos i t i ons 2.1 and 2.2 and a d i r e c t - a p p l i c a t i o n of 151, Theorem 2.3.

REFERENCES

[ I ] ANDRUN&IEVIC, V.A. Radicals i n a s s o c i a t i v e r i n g s I , Amer. Math. Soc. Tr'rrmsZ. S e r . 2 , 52 ( l 966 ) , 95-1 28

Dow

nloa

ded

by [

UQ

Lib

rary

] at

20:

45 2

0 N

ovem

ber

2014

148 GROENEWALD AND HEYMAN

A N D R U N ~ I E V I C , V. A. and RJABUHIN, Yu.M. Rings w i t h o u t n i l p o t e n t e l ement s and comple te ly s imple i d e a l s , Sovie t Math. Dok. 9 ( l 9 6 8 ) , 565-568

DIVINSKY, N . J . Rings and RadicaZs, Univ. o f Toron to P r e s s , 1965

GROENEWALD, N . J . S t r o n g l y pr ime group r i n g s , &uaestiones Mathematicae 3 ( I 9 7 9 ) , 241-247

GROENEWALD, N . J . and H.J. SCHUTTE. C e r t a i n c l a s s e s o f i d e a l s i n group r i n g s , Corn. i n Algebra V . 8 No. l (1 98O), 87-104

HANDELMAN, D. and J . LAWRENCE. S t r o n g l y pr ime r i n g s , Trans. Amer. Math. Soc. 21 1 ( l 9 7 5 ) , 209-229

HERSTEIN, I. N. ~Joncomutatiue Rings, Carus Math. Monographs No. 15 , Math. Assoc. Amer., 1968

HEYMAN, G.A.P. R a d i c a l s i n a s s o c i a t i v e r i n g s , T h e s i s , U n i v e r s i t y of B loemfon te in , South A f r i c a (1974)

HEYMAN, G.A.P. and C. ROOS. E s s e n t i a l e x t e n s i o n s i n r a d i c a l t h e o r y f o r r i n g s , J. AustraZ. Math. Soc. 23 ( S e r i e s A) ( l 9 7 7 ) , 340-347

KREMPA, J. On Semi-group Rings , BUZZ. Acad. PoZon. Sc i . , 25 ( l 9 7 7 ) , 225-231

McCOY, N.H. Completely pr ime and comple te ly semi-prime i d e a l s , CoZZ. Math. Soc. J . BoZyai, 6 , R ings , modules and r a d i c a l s , North Hol l and , 1973, 147-152

[12] NICHOLSON, W.K. and J . F . WATTERS. The s t r o n g l y pr ime r a d i c a l , Proc. Amer. Math. Soc., 76 (1979) , 235-240

[13] PASSMAN, D. S. The Algebraic Structure of Group Rings, A Wiley I n t e r s c i e n c e P u b l i c a t i o n , John Wiley and Sons , New York, 1977

[14] RUBIN, R.A. A b s o l u t e l y t o r s i o n f r e e r i n g s , Pac. J. Math. 46 (1 9 7 3 ) , 503-51 4

Received: November 1979

Dow

nloa

ded

by [

UQ

Lib

rary

] at

20:

45 2

0 N

ovem

ber

2014