NEW ZEALAND JOURNAL OF MATHEMATICS Volume 22 (1993), 93-98

STRONGLY PRIME GROUP RINGS

R.K. S h a r m a a n d J.B. S r i v a s t a v a

(Received May 1992)

Abstract. It is proved that if R is any strongly prime ring and G is any one-relator group then the group ring RG is strongly prime, if G is FC-solvable then R G is strongly prime if and only if R is strongly prime and G contains no nontrivial locally finite normal subgroups; if G is a group having a finite normal series with F C - hypercentral factors then R G is strongly prime if and only if R is strongly prime and the locally finite radical of G is trivial; if K is a field then G can be imbedded into an infinite simple group G* such that K G * is strongly prime. Some examples are also given.

1. IntroductionA ring R is said to be right strongly prime (henceforth called strongly prime) if

every nonzero two sided ideal of R contains a finite subset whose right annihilator is zero. These rings were first systematically studied by Handelman and Lawrence [8], They raised the problem of characterizing strongly prime group rings ([8], Question 3, p. 222) and conjectured the following:

Conjecture. ([8], Pages 215 and 222) The group ring RG is strongly prime if and only if R is strongly prime and G contains no nontrivial locally finite normal subgroups.

The main aim of this paper is to settle in affirmative this conjecture for the class of one relator groups, FC-solvable groups which include solvable by finite groups as a particular case and the groups having finite normal series with factors F C - hypercentral. When R — K is a field, we determine some more groups G including certain wreath products for which the corresponding group algebra K G is strongly prime.

Domains (not necessarily commutative) and prime Goldie rings are strongly prime. It is proved ([8], IILl.(a)) that if the group ring RG is strongly prime then R is strongly prime and G contains no nontrivial locally finite normal sub­groups. Hence proving the conjecture in affirmative simply means proving the converse of this theorem. Strongly prime rings are prime. Connell [3, Theorem 8] has shown that RG is prime if and only if R is prime and G contains no nontriv­ial finite normal subgroups. For a study of strongly prime ideals, we refer to the beautiful paper by Ferrero et al. [5].

2. Preliminary LemmasThroughout this paper a ring will mean an associative ring with identity. For

a normal subgroup H of a group G and for an ideal I of a ring R, we shall write H < G and I < R, respectively. X is a normal subgroup of Y or an ideal of Y in the notation X < Y will be clear from the context. For a group G , let L(G)

1991 A M S Mathematics Subject Classification-. 16N99, 16S34

94 R.K. SHARMA and J.B. SRIVASTAVA

denote the locally finite radical of G, that is, L (G ) is the unique maximal locally finite normal subgroup of G. Clearly L(G) is a characteristic subgroup of G and is the union of all locally finite normal subgroups of G.

The following two lemmas are straightforward.

Lemma 2.1. Let R be a ring, G a group and H < G such that I fl R H ^ 0 for every 0 ^ I < RG. Then RG is strongly prime if R H is strongly prime.

Lemma 2.2. Let H be a subnormal subgroup of a group G. Then L{G) = (1) implies L(H) = (1).

A group G is said to be F C if every element of G has only finitely many con­jugates in G. In a periodic FC-group every finite subset is contained in a finite normal subgroup.

The following lemma plays a crucial role in our further work.

Lemma 2.3. Let R be a ring, G a group, H < G such that L(G) = (1) and G /H is a periodic FC-group. Then RG is strongly prime if R H is strongly prime.

Proof. If W < G, then R W is prime by [3, Theorem 8]. This is because L (W ) = (1) by Lemma 2.2 and R is prime, since R H is strongly prime.

Let 0 7£ I < RG and 0 ^ a = ri9i e Let N /H be the smallest normal subgroup of G /H containing the cosets {giH , g2H , . . . ,gnH }. Then \N : H\ < oo, since N /H is finite as G /H is a periodic FC-group. Clearly g\,g2, ■ ■ ■ ,gn belong to N and hence a = Y ^ = i ri9i e ̂^ Thus I fl R N ^ 0.

We see that R N /R H is a finite normalizing extension of rings such that R N is R H -free and both R N and R H are prime. Hence by [9, Theorem 4.16], we have ( / fl RN) fl R H = I fl R H ^ 0. The result now follows from Lemma 2.1.

Groenewald [7, Theorem 2.1] proved that RG is strongly prime if R is strongly prime and G is a unique product group. We shall need later the following.

Lemma 2.4. [7, Theorem 2.2] Let R be a ring and let H < G such that G /H is a right ordered group. Then RG is strongly prime if R H is strongly prime.

3. One Relator GroupsOne relator groups have been extensively studied recently. A group is said to be

locally indicable if each of its nontrivial finitely generated subgroups has the infinite cyclic subgroup as homomorphic image. Howie [10, Corollary 4.3] proved that all torsion free one relator groups are locally indicable. Burns and Hale [2, Corollary 2] proved that locally indicable groups are right orderable (RO ). Combining the two, we get that every torsion free one relator group is an RO group. Also, i?0-groups are unique product (UP) groups. This settles the question raised by Lichtman [12, §4]. We refer to [6] and [13] as standard references for one relator groups.

The main result in this section is to prove the following.

Theorem 3.1. The group ring RG is strongly prime if R is strongly prime, G is any one relator group with at least two generators.

STRONGLY PRIME GROUP RINGS 95

Proof. Let G = (X ;S ) with |X| > 2 be a one relator group and R be any strongly prime ring. If G is torsion free then it is an RO-group as noted above. Also, by Lemma 2.4, RG is strongly prime.

Now, suppose that G is not torsion free. Then G = (X ; S ) , S = T n, n > 2, T is not a proper power and T is a cyclically reduced word. By [6 , Theorem 2] G contains a torsion free normal subgroup Go with |G : Go I < 00 •

Let H be the subgroup of G generated by all torsion elements of G. Then by [6 , Theorem 1] H is the free product of all conjugates of the cyclic subgroup generated by the image of T. Clearly G /H = (X-,T) is a torsion free one relator group. Thus G /H is an .RO-group.

Now Go H H is a torsion free subgroup of H which is a free product of finite cyclic groups, hence by Kurosh subgroup Theorem [13, IV. 1.10], Go H H — F is a free group.

Further G q /F = Gq/Gq fl H = G q H /H and G q H /H is a subgroup of an RO- group G /H , hence G o /F is an .RO-group. Also F is an .RO-group, since it is free. Thus Go is an .RO-group [14, 13.1.5] and RGq is strongly prime by Lemma 2.4.

Finally it is not difficult to see that the locally finite radical of G is trivial, that is, L(G) = (1). Now by Lemma 2.3, RG is strongly prime, since RGq is strongly prime and G/Go is finite.

4. FC-Solvable and FC-H ypercentral GroupsFollowing Duguid and McLain [4], a group G is called FC-solvable if it has a

finite subnormal series,

1 = Ho <! H\ H 2 55 • • • ^ H n -i — Hn — G

such that Hi- 1 < Hi and H i/H i-i is an FG-group for i = 1 ,2 ,... , n. For an arbitrary group G, the F C - centre A(G) = {x € G | |G : Cg{x)\ < 00} and the torsion part of A(G) is A + (G) = {x € A(G) | x has finite order}. A(G) and A + (G) are characteristic subgroups of G, A (G )/A + (G) is torsion free abelian, A + (G) is locally finite and A + (G) is the union of all finite normal subgroups of G ([14], Chapter 4).

Thus G is an FC-group if and only if G = A(G). Also for any group L(G) = (1) implies A +(G) = (1) and A(G) is torsion free abelian.

Theorem 4.1. Let R be a ring and G an FC-solvable group. Then RG is strongly prime if and only if R is strongly prime and L(G) = (1).

Proof. If RG is strongly prime, then the conclusion follows from [8 , IILl.(a)].

Conversely suppose that R is strongly prime and L(G) = (1). Since G is F C - solvable, it has a finite subnormal series 1 = Ho <! Hi < H 2 < . . . < Hn_ 1 < Hn = G such that Hi- 1 < Hi and H i/H i- 1 is an FC-group for i = 1 ,2 ,... ,n.

We shall prove, by induction, that RHk is strongly prime for k = 1 ,2 ,... ,n. First of all, H\ is subnormal in G and L(G) = (1), hence by Lemma 2.2, L(H\) =

(1). Also H\ is an FC-group, so Hi is torsion free abelian. Thus RHi is strongly prime by [8 , IILl.(b)].

96 R.K. SHARMA and J.B. SRIVASTAVA

Assume that RHi is strongly prime for i = 1 ,2 ,... ,k. Now H k+i /H k is an FC-group, hence its torsion subgroup is given by (H k+i /H k)+ = \[HkfH k ̂ where y/Jh = {z € H k+1 | xm e H k for some m > 1}. Clearly v'tffc < H k+1, s/W kfH k is a periodic FC-group, and H k+\/y/TTk is torsion free abelian.

Further L(\fHk) — (1) by Lemma 2.2, since y/Hk is subnormal in G and L(G) = (1). Now R H k is strongly prime, \JTTk/Hk is a periodic FC-group, so by Lemma 2.3, R^/Hk is strongly prime.

Next Ry/Tlk is strongly prime, H k+i/y/TTk is torsion free abelian, hence ordered and therefore by Lemma 2.4, R H k+\ is strongly prime. The induction is complete and RG = RHn is strongly prime, as desired.

Corollary 4.2. Let R be a ring and G a nilpotent group. Then RG is strongly prime if and only if R is strongly prime and G is torsion free.

Corollary 4.3. Let R be a ring and G a solvable by finite group. Then RG is strongly prime if and only if R is strongly prime and L(G) = (1).

The FC'-chain of a group G is defined by A i(G ) = A(G ), Aa+i(G?)/Aa(G') = A (G /A \ (G )) for an ordinal A, and A fX(G) = Ua<mAa(G) if p is a limit ordinal. The group G is called hyper-A or FC-hypercentral if G = A a(G) for some ordinala. It is easy to see that G is hyper-A if and only if A ( G/ H) is nontrivial for every proper normal subgroup H of G. The following theorem can be proved on the lines similar to Theorem 4.1, using transfinite induction.

Theorem 4.4. Let 1 = Go < Gi < < G a = G for some ordinal a be a series of normal subgroups of G, where G^ = ^\<^G\ for a limit ordinal and G \+i/G \ is an FC-group for every A. Then RG is strongly prime if and only if R is strongly prime and L(G) = (1).

Corollary 4.5. Let R be a ring and G a hyper-A group. Then RG is strongly prime if and only if R is strongly prime and L(G) = (1).

Corollary 4.6. Let G be a group having a finite normal series 1 = Go <! G\ <. . . < Gn = G such that each factor G i /G i-i is hyper-A. Then RG is strongly prime if and only if R is strongly prime and L(G) = (1).

Proof. Let R be strongly prime and L(G) = (1). Then L(G\) = (1) by Lemma2.2 and G\ is hyper-A, so RG\ is strongly prime by Corollary 4.5. Next assume that RGi is strongly prime for some i > 1. Since Gi+ i/G i is hyper-A, there exists a series Gi — Ho < H\ < < Ha = G l+\ for some ordinal a such that each H\ < Gi+1 and H\+ i/H \ is an FC-group. So we get that RGi+ 1 is strongly prime. Thus RG = RGn is strongly prime.

5. Group AlgebrasIn this section we take R = K to be a field. In [1] strongly prime group algebras

over fields are studied. Many authors in many contexts have obtained sporadic results on strongly prime group algebras using known intersection theorems and the existing related group ring results, (cf. [14]). We strictly adhere to certain

STRONGLY PRIME GROUP RINGS 97

comments, remarks and basically some new examples to show that the study of strongly prime group algebras is very much desirable.

First we observe that for a nilpotent group G, the group algebra K G is strongly prime if and only if G is torsion free if and only if K G is a domain. However, if G is a polycyclic by finite group then K G is Noetherian and so K G is strongly prime if and only if K G is prime. When K G is strongly prime, at times K G is a domain or very close to a domain, however sometimes it is just prime and far from being a domain. We show that any group can be embedded in an infinite simple group whose group algebra is strongly prime. Also we conclude via wreath products that any infinite group can occur as a homomorphic image of a group whose group algebra is strongly prime.

Proposition 5.1. Let K be a field and G a group. Then G can be embedded in an infinite simple group G* such that K G * is strongly prime.

Proof. Let G\ = G x where C00 = (x) is infinite cyclic. By [14, 9.4.4] G\ can be embedded into an algebraically closed group G* which is infinite simple. By [14, 9.4.5, 9.4.6], the augmentation idal u>(KG*) is the only nonzero proper ideal in K G *. Now lj(K G *) contains x — l and Ann^G*{x — 1) = 0, since x has infinite order. Thus by definition KG * is strongly prime.

On the contrary by [14, 9.4.9, 9.4.10] if G is a universal locally finite group, then G is infinite simple and the augmentation ideal uj(KG) is the only nonzero proper ideal of K G , but K G is not strongly prime.

Our next result gives a rich source of examples.

Proposition 5.2. Let G — A \B be the wreath product of groups A and B with A 7̂ (1) and B infinite. If H = YlbeB Af, is the base group such that K H is strongly prime, then K G is also strongly prime.

Proof. By [14, 9.2.7], I fl K H ^ 0 for any nonzero ideal I of K G . The result follows by Lemma 2.1.

Remark 5.3. Let A be a non trivial torsion free solvable group and B be any infinite group. Then K A is a domain by [11]. Thus if G = A \B and K is a field, then K G is strongly prime.

Acknowledgements. The authors wish to express their sincere thanks to the referee for valuable suggestions.

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R .K . SharmaIndian Institute of TechnologyKharagpur 721302INDIA

J.B. SrivastavaIndian Institute of TechnologyNew Delhi 110016INDIA