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∗-PRIME GROUP RINGS 1. Introduction Let R Be an Associative December 15, 2009 14:39 WSPC/171-JAA 00364 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). 797 December 15, 2009 14:39 WSPC/171-JAA 00364 798 K. Joshi, R. K. Sharma & J. B. Srivastava 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 sub- group H of G,Λ(G)H/H ⊆ Λ(G/H)andΛ+(G)H/H ⊆ Λ+(G/H).
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