∗-PRIME GROUP RINGS 1. Introduction Let R Be an Associative

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Journal of Algebra and Its Applications Vol. 8, No. 6 (2009) 797–803 c World Scientiﬁc Publishing Company

∗-PRIME GROUP RINGS

KANCHAN JOSHI∗,R.K.SHARMA† and J. B. SRIVASTAVA ‡ Department of Mathematics Indian Institute of Technology Delhi New Delhi, India ∗[email protected] †[email protected] ‡[email protected]

Received 20 January 2009 Accepted 17 March 2009

Communicated by M. Ferrero

In this paper we characterize ∗-prime group rings. We prove that the group ring RG of the group G over the ring R is ∗-prime if and only if R is ∗-prime and Λ+(G) = (1). In the process we obtain more examples of group rings which are ∗-prime but not strongly prime.

Keywords: Strongly prime; ∗-prime; prime; group rings.

Mathematics Subject Classiﬁcation 2000: 16S34, 16N60, 20C07

1. Introduction

Let R be an associative ring with identity 1 =0and G be a group. It is well known [1] that the group ring RG of the group G over the ring R is prime if and only if R is prime and the FC subgroup ∆(G) is torsion free abelian or equivalently its torsion subgroup ∆+(G) = (1). AringR is said to be (right) strongly prime if for every 0 = r ∈ R,there exists a finite subset X of R such that rXt =0,t∈ R ⇒ t = 0. Left strongly prime rings are defined analogously. These were first studied by Handelman and Lawrence [2]. Clearly, strongly prime rings are prime. For group rings they proved that if RG is strongly prime then R is strongly prime and G has no non-trivial locally finite normal subgroup, that is, the locally finite radical L(G) = (1). Also, they conjectured that if R is strongly prime and G is a group such that L(G) = (1), then RG is strongly prime. Sharma and Srivastava [5] have settled the conjecture

∗The author acknowledges the ﬁnancial support of the Council of Scientiﬁc and Industrial Research (India).

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in affirmative for various classes of groups which include one relator groups and FC-solvable groups. For a group G,Λ+(G) denotes the torsion subgroup of the locally FC subgroup Λ(G)ofG. In view of the containment ∆+(G) ⊆ Λ+(G) ⊆ L(G), a detailed study was carried out in our work [3] about the algebraic structure of group algebras over fields for groups G with Λ+(G) = (1). In the process, we introduced ∗-prime algebras. An algebra A is called (right) ∗-prime if for every 0 = a ∈ A,thereexists a finitely generated subalgebra B of A such that aBt =0,t ∈ A, implies t =0.We proved that the group algebra KG of the group G over the field K is ∗-prime if and only if Λ+(G) = (1). We study here ∗-prime group rings over arbitrary rings. AringR is said to be (right) ∗-prime if for every 0 = r ∈ R, there exists a finitely generated subring S of R such that rSt =0,t∈ R, implies t =0. Left ∗-prime rings are defined analogously. It is not difficult to see that an algebra is ∗-prime as an algebra if and only if it is ∗-prime as a ring. Hence the two definitions are compatible. Clearly, ∗-prime rings are prime and strongly prime rings are ∗-prime. In our work [3], we have given a large number of examples to show that the concept of ∗-prime lies strictly between that of prime and strongly prime. For group algebras we have observed that left and right ∗-prime notions coincide. However, this is not the case with group rings over arbitrary rings. In this paper we give the characterization of ∗-prime group rings. We show that the group ring RG is (right) ∗-prime if and only if R is (right) ∗-prime and Λ+(G)=(1). As a consequence, we get a large number of examples of group rings which are ∗-prime but not strongly prime.

2. Lemmas In this section, we give two lemmas required for proving our main result in the next section. The ﬁrst lemma deals with group rings of unique product groups. AgroupG is said to be a unique product group if given any two non-empty subsets A and B of G, there exists at least one element x ∈ G that has a unique representation of the form x = ab,witha ∈ A and b ∈ B. Unique product groups are always torsion-free. Ordered groups are unique product groups. It is known that torsion-free abelian groups are ordered and hence unique product groups. For further details see [4, Chapter 13].

Lemma 2.1. Let H be a normal subgroup of a group G such that G/H is a unique product group and let R be a ring. Then RH is ∗-prime implies RG is ∗-prime. In particular, if R is a ∗-prime ring and G is a unique product group, then the group ring RG is ∗-prime. Proof. ∈ n ∈ ∈ Let 0 = α RG.Thenα = i=1 αigi,where0= αi RH, gi T , where T denotes a transversal of H in G.AsRH is ∗-prime, for each αi there exists a ﬁnitely generated subring Si of RH such that αiSiβ =0,β ∈ RH ⇒ β =0, December 15, 2009 14:39 WSPC/171-JAA 00364

∗-Prime Group Rings 799 { n n gj } i =1, 2,...,n.LetS be the subring of RG generated by i=1 j=1 Si ,where gj −1 −1 Si = gj Sigj.ThenS is a finitely generated subring of RG as each gj Sigj is finitely generated. In fact, normality of H implies that S is a finitely generated subring of RH. ∈ m Now let if possible αSβ =0,forsome0= β RG.Thenβ = i=1 βixi, for some 0 = βi ∈ RH, xi ∈ T .LetA = {Hg1,Hg2,...,Hgn} and B = {Hx1,Hx2,...,Hxm}.ThenasG/H is a unique product group, there exists 1 ≤ l ≤ n,1≤ t ≤ m such that Hgl.Hxt = Hglxt is uniquely represented. Now   n m n m −1 −1 g ⇒   ⇒ gi i αSβ =0 αigi S βjxj =0 (αiS βj gixj)=0 i=1 j=1 i=1 j=1

−1 −1 −1 g −1 g gi i ⊆ gi i ∈ ≤ ≤ ≤ Also Supp(αiS βj gixj) Hgixj since αiS βj RH,for1 i n,1 −1 −1 g ≤ gl l j m.AsHglxt is uniquely represented, we obtain αlS βt = 0. This implies −1 −1 −1 gl g gl g gl l ⊆ l αl(Sl ) βt =0sinceSl S. As a consequence, αlSlβt = 0 which implies −1 g l ∈ ⇒ βt = 0. This is a contradiction because βt = 0. Hence αSβ =0,β RG β =0. Thus RG is ∗-prime. The particular case follows by taking H =(1).

Let G be an arbitrary group. Then its FC subgroup is deﬁned as ∆(G)={x ∈ + G ||G : CG(x)| < ∞} and its torsion subgroup ∆ (G)={x ∈ ∆(G) | o(x) < ∞}. It is well known that ∆(G)and∆+(G) are characteristic subgroups of G and ∆(G)/∆+(G) is torsion-free abelian. We also need the following lemma for group rings over arbitrary rings. For group algebras over ﬁelds this is a particular case of [4, Lemma 4.2.4].

Lemma 2.2. Let R be a ring and G be a group. Suppose αgβ =0∀ g ∈ G and for

some α, β ∈ RG.Thenπ∆(G)(α)β =0.

Proof. ∈ Suppose π∆(G)(α)β =0.Letg Supp(π∆(G)(α)β) and let Supp(π∆(G)(α)) = { } n | | ∞ ∈ x1,x2,...,xn . Deﬁne H = i=1 CG(xi). Then G : H < as each xi ∆(G). Let α = π∆(G)(α)+α0, where Supp(α0)={y1,y2,...,yt}⊆G\∆(G). Let { } −1 Supp (β)= z1,z2,...,zm .Ifyi is conjugate to gzj for some j, then there exists ∈ −1 −1 ∈ uij G such that uij yiuij = gzj . Suppose h H.Then 0=h−1αhβ −1 −1 = h π∆(G)(α)hβ + h α0hβ −1 = π∆(G)(α)β + h α0hβ. −1 −1 Now as g ∈ Supp(π∆(G)(α)β), g ∈ Supp(h α0hβ). Thus g = h yihzj for some ≤ ≤ ≤ ≤ −1 −1 −1 ⇒ −1 ∈ ⇒ 1 i t and 1 j m.Then h yih = gzj = uij yiuij huij CG(yi) ∈ ⊆ | | ∞ h CG(yi)uij .ThusH i,j CG(yi)uij .As G : H < , G = i,j,k CG(yi)uij tk, December 15, 2009 14:39 WSPC/171-JAA 00364

800 K. Joshi, R. K. Sharma & J. B. Srivastava

where tk belong to the transversal for H in G. By [4, Lemma 4.2.1], |G : CG(yi)| < ∞ for some i, which is a contradiction as yi ∈ ∆(G). Hence π∆(G)(α)β =0.

3. Main Result In this section, we give the main result of this paper. In order to prove this theorem we need the following details about the locally FC subgroup Λ(G)ofG. Let H be a subgroup of G.Then|G : H| = l.f.,wherel.f. stands for locally finite, if for every finitely generated subgroup L of G, |L : L ∩ H| < ∞. Define + Λ(G)={x ∈ G ||G : CG(x)| = l.f} and Λ (G)={x ∈ Λ(G) | o(x) < ∞} = {x ∈ G | x has finitely many conjugates under every finitely generated subgroup of G and o(x) < ∞}.Λ(G)andΛ+(G) are characteristic subgroups of G.Furthermore, Λ+(G) is locally finite and Λ(G)/Λ+(G) is torsion-free abelian. For a normal subgroup H of G,Λ(G)H/H ⊆ Λ(G/H)andΛ+(G)H/H ⊆ Λ+(G/H). For details see [4, Chapter 8]. Now we derive our main theorem.

Theorem 3.1. Let R be a ring and G be a group. Then RG is ∗-prime if and only if R is ∗-prime and Λ+(G)=(1).

Proof. Let R be ∗-prime and Λ+(G) = (1). Let 0 = α ∈ RG. Without loss of

generality, we can assume that 1 ∈ Supp(α). Then 0 = πΛ(G)(α) ∈ R[Λ(G)] which is ∗-prime by Lemma 2.1 as Λ(G) is torsion-free abelian. Hence there exists a ﬁnitely

generated subring, say, T = α1,...,αn of R[Λ(G)] such that πΛ(G)(α)Tγ =0, ∈ ⇒ ∪ n ∈ \ γ R[Λ(G)] γ =0.LetX = Supp (α) ( i=1 Supp(αi)). For every g G Λ(G), | | ∞ there exists a ﬁnitely generated subgroup Hg of G such that Hg : CHg (g) = . Let G0 be the subgroup of G generated by {X, Hg | g ∈ X (G\Λ(G))}.ThenG0 is a ﬁnitely generated subgroup of G such that

X ∩ Λ(G) ⊆ ∆(G0)

X ∩ (G\Λ(G)) ⊆ G0\∆(G0). ∈ ∈ Suppose αi = j rij gij ,0= rij R and gij G.LetR0 be the subring of R generated by {rij | i, j}.ThenR0G0 is a ﬁnitely generated subring of RG.

Claim. αR0G0β =0,β ∈ RG ⇒ β =0.

Suppose αR0G0β =0andβ = 0. Without loss of generality, we can assume 1 ∈ Supp(β). Let β = β0 + β1, where Supp(β0) ⊆ G0 and Supp(β1) ∩ G0 = ∅.Now αR0G0β =0⇒ αR0G0β0 + αR0G0β1 =0⇒ αR0G0β0 =0.AsT ⊆ R0G0,we ∀ ∈ obtain αT xβ0 =0 x G0. Using Lemma 2.2 we obtain π∆(G0)(αT )β0 =0.As Supp(αi) ⊆ Λ(G) ∩ X ⊆ ∆(G0) for every i =1, 2,...,n, T ⊆ R[∆(G0)]. Thus by ⊆ [4, Lemma 1.1.2], π∆(G0)(α)Tβ0 = 0. Now as Supp(α) X, π∆(G0)(α)=πΛ(G)(α). Hence πΛ(G)(α)Tβ0 =0⇒ πΛ(G)(α)TπΛ(G)(β0) = 0. This is a contradiction as πΛ(G)(β0) = 0 because 1 ∈ Supp(β0). December 15, 2009 14:39 WSPC/171-JAA 00364

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Hence β =0andRG is ∗-prime. Conversely, suppose RG is ∗-prime. Let 0 = r ∈ R.AsRG is ∗-prime, there ⇒ exists a finitely generated subring S = s1,s2,...,sn of RG such that rSt =0 ∈ t =0,t RG.Letsi = j rij xij and let R0 be the subring of R generated by {rij |i, j}.ThenrR0r =0,r ∈ R ⇒ rSr =0⇒ r = 0. Hence we obtain that R is ∗-prime. Suppose if possible Λ+(G) =(1).Let g ∈ Λ+(G), g =1.Then0 = g − 1 ∈ RG. As RG is ∗-prime, there exists a finitely generated subring S = β1,β2,...,βn of − ∈ RG such that (g 1)St =0,t RG, implies t = 0. Without loss of generality, ∈ n −1 | ∈ we can assume g S.LetH = g, i=1 Supp(βi) .LetN = h gh h H . Now as g ∈ Λ+(G)andH is a finitely generated subgroup of G, g has finitely many conjugates under H.ThusN is a finitely generated subgroup of Λ+(G), which is locally finite. Hence N is finite. Also N is a normal subgroup of H.Let ˆ ˆ ˆ ∈ − ˆ N = y∈N y. Clearly, S.N = N.S and since g N,(g 1)N =0.Thus (g − 1)S.Nˆ =(g − 1)N.Sˆ =0. However, Nˆ = 0. This contradicts the fact (g − 1)St =0⇒ t =0. Hence Λ+(G) = (1).

Corollary 3.2. Let R be a ring and H be a normal subgroup of G such that Λ+(G/H)=(1).ThenRH is ∗-prime implies RG is ∗-prime.

Proof. RH is ∗-prime implies R is ∗-prime and Λ+(H) = (1). Also since Λ+(G)H/H ⊆ Λ+(G/H) = (1), Λ+(G) ⊆ H ⇒ Λ+(G) ⊆ Λ+(H) = (1). Hence by the above theorem RG is ∗-prime.

Using Theorem 3.1 and characterization of prime group rings, we obtain the following important result.

Theorem 3.3. Let G be a group such that Λ+(G)=(1)and let R be a ∗-prime ring. Then RG is ∗-prime if and only if it is prime.

Proof. Let x ∈ Λ+(G). Then x−1g−1xg =(x, g) ∈ Λ+(G) ∩ G ⊆ Λ+(G)= (1) ⇒ xg = gx ∀ g ∈ G ⇒ x ∈Z(G) = center of G ⊆ ∆(G). Hence Λ+(G)= ∆+(G). Thus by Theorem 3.1 and [1], we obtain RG is ∗-prime if and only if it is prime.

4. Examples In this section, we give a large number of examples of group rings RG of a group G over a ring R which are ∗-prime but not strongly prime. These follow as a consequence of results proved in this section. Here R need not be even commutative. We use Theorem 3.1 which gives that RG is ∗-prime if and only if R is ∗-prime and Λ+(G) = (1). December 15, 2009 14:39 WSPC/171-JAA 00364

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Standard wreath product of groups is a rich source of examples in our study. Let G1 and G2 be any two non-trivial groups. Then G = G1 G2, the standard wreath product ofG1 by G2, is the semidirect product G = W θ G2 where the b b { | ∈ 1} base group W = b∈G2 G1 (weak direct product), G1 = [a, b] a G with b ∼ multiplication defined by [a1,b][a2,b]=[a1a2,b]. Thus G1 = G1 ∀ b ∈ G2.Every b bc c ∈ G2 induces an isomorphism of G1 → G1 by [a, b] → [a, bc] and hence induces an automorphism, say, θ,ofW . This gives an action of G2 on W . Thus the product ∈ ∈ in G is defined by (w1,b1)(w2,b2)=(w1θb1 (w2),b1b2), w1, w2 W , b1, b2 G2. We have the following results which were known to be true when R is a field.

Theorem 4.1. Let G = G1 G2,G1 =(1) and G2 inﬁnite and let R be a ring. Then ∆(G)=(1)and I ∩ RH =(0) for every nonzero ideal I of RG, where H is the base group of G.

Proof. Follows exactly on the same lines as the proof of [4, Lemma 9.2.7].

Lemma 4.2. Let H be a normal subgroup of G such that I ∩ RH =(0) for every nonzero ideal I of RG.ThenRH is ∗-prime implies RG is ∗-prime.

Proof. Proof is on the same lines as the proof of [3, Lemma 3.1]

As a consequence we have the following.

Corollary 4.3. Let G = G1 G2,whereG1 =(1) , G2 is an infinite group and let R be a ring. Let H bethebasegroupofG.ThenRH is ∗-prime implies RG is ∗-prime. Further, if G2 is infinite locally finite, then RH is ∗-prime if and only if RG is ∗-prime.

Proof. The first part follows from Theorem 4.1 and Lemma 4.2. Suppose G2 is ∼ locally finite, then G/H = G2 is locally finite. It is not difficult to see that in this case Λ+(H) ⊆ Λ+(G). If RG is ∗-prime, then R is ∗-prime and Λ+(G) = (1). Thus Λ+(H)=(1)andR is ∗-prime ⇒ RH is ∗-prime.

Example 4.4. Let R be a ∗-prime ring and let G = G1 G2, G1 is a non-trivial locally finite group and G2 is an infinite group which is not locally finite. Using [3, Theorem 2.4], we obtain Λ(G) = (1). Then by Theorem 3.1, RG is ∗-prime. But the base group is a non-trivial locally finite normal subgroup of G. Hence RG is not strongly prime. In particular, let G = Cp C∞ and let R be a prime noetherian ring. Then R is strongly prime [2] hence ∗-prime and Λ(G) = (1). Thus RG is ∗-prime but not strongly prime. In this example Cp can be replaced by any non-trivial locally finite group. In Example 4.4, in order to prove RG is not strongly prime, we used that G has a non-trivial locally finite normal subgroup. In the next example we take R to December 15, 2009 14:39 WSPC/171-JAA 00364

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be any ∗-prime ring which is not strongly prime. Hence RG will not be strongly prime.

Example 4.5. Let R be any ∗-prime ring which is not strongly prime and let G be a group with Λ+(G) = (1). By Theorem 3.1, RG is ∗-prime. However, RG is not strongly prime by [2].

References [1] I. G. Connell, On the group ring, Canad. J. Math. 15 (1963) 650–685. [2] D. Handelman and J. Lawrence, Strongly prime rings, Trans. Amer. Math. Soc. 211 (1975) 209–223. [3] K. Joshi, R. K. Sharma and J. B. Srivastava, ∗-Prime group algebras, Comm. Alg. 35(11) (2007) 3673–3682. [4]D.S.Passman,The Algebraic Structure of Group Rings (Wiley-Interscience, New York, 1977). [5] R. K. Sharma and J. B. Srivastava, Strongly prime group rings, New Zealand J. Math. 23 (1994) 93–98.