CUBO A Mathematical Journal Vol.14, No¯ 01, (49–54). March 2012 Units in Abelian Group Algebras Over Direct Products of Indecomposable Rings Peter Danchev 13, General Kutuzov Str. 4003 Plovdiv, Bulgaria, email: pvdanchev@yahoo.com ABSTRACT Let R be a commutative unitary ring of prime characteristic p which is a direct product of indecomposable subrings and let G be a multiplicative Abelian group such that G0/Gp is finite. We characterize the isomorphism class of the unit group U(RG) of the group algebra RG. This strengthens recent results due to Mollov-Nachev (Commun. Algebra, 2006) and Danchev (Studia Babes Bolyai - Mat., 2011). RESUMEN Sea R un anillo conmutativo y unitario de caracteŕıstica prima p, que es producto directo de subanillos indescomponibles y sea G un grupo multiplicativo y abeliano tal que G0/Gp es finito. Caracterizamos las clases de isomorfismo del grupo unitario U(RG) del álgebra del grupo RG. Estos fuertes y recientes resultados se deben a Mollov-Nachev (Commun. Algebra, 2006) and Danchev (Studia Babes Bolyai - Mat., 2011). Keywords and Phrases: groups, rings, group rings, indecomposable rings, units, direct decom- positions, isomorphisms. 2010 AMS Mathematics Subject Classification: 16S34, 16U60, 20K21. 50 Peter Danchev CUBO 14, 1 (2012) 1 Introduction Throughout the current paper, suppose R is a commutative unitary (i.e., with identity) ring of prime characteristic p and suppose G is a multiplicative Abelian group as is the custom when discussing group rings. For such R and G, we denote by RG the group ring of G over R with unit group U(RG), normalized subgroup V(RG) of units (with augmentation 1) and its idempotent subgroup Id(RG). Note that the decomposition U(RG) = V(RG)×U(R) is valid, where U(R) is the unit group of R. As usual, G0 is the maximal torsion subgroup of G with p-torsion component Gp, and S(RG) = Vp(RG) is the p-torsion component of V(RG). Besides, for any natural number n, ζn denotes the primitive nth root of unity and R[ζn] is the free R-module, generated algebraically as a ring by ζn, with dimension [R[ζn] : R]. As it is well-known, a ring is said to be indecomposable if it cannot be decomposed into a direct sum of two or more non-trivial subrings (ideals), that is, this ring possesses only the trivial idempotents 0, 1. The algebraic structures of V(RG) and U(RG) have been very intensively explored in the past twenty years (see, e.g., [K]). In this aspect, some isomorphism description results were obtained in [Da] and [MN], respectively. The purpose of this work is to improve considerably one of the central achievements in the second citation by giving a more direct and conceptual proof (some of parts of the proof of the corresponding result in [MN] are unnecessary intricated). Likewise, we generalize the main result in [Dg] to a ring which is an arbitrary direct product of indecomposable rings. Notice that our method suggested below gives a new perspective for establishing some other results of this form, because it leads the general case to the p-mixed one. II. Main Results As noted above, Mollov and Nachev obtained in ([MN], Theorem 5.8) the following statement. Theorem (2006). Let R be a commutative indecomposable ring with identity of prime characteristic p and let G be a splitting Abelian group. Suppose that G0/Gp is a finite group of exponent n and n ∈ U(R). Then U(RG) ∼= ∐ d/n ∐ λ(d) U(R[ζd]) × ∐ b G/G0 × ∐ d/n ∐ λ(d) S(R[ζd](Gp × G/G0)) where λ(d) = (G0/Gp)(d) [R[ζd]:R] , with (G0/Gp)(d) the number of elements of G0/Gp of order d, and b = ∑ d/n λ(d). Note that since char(R) = p is a prime integer, it is self-evident that exp(G0/Gp) inverts in R, so that the condition n ∈ U(R) is always fulfilled and hence it is a superfluously stated in the theorem. CUBO 14, 1 (2012) Units in Abelian Group Algebras Over Direct Products . . . 51 In [Dg] we dropped the limitation that G is a splitting group. Specifically, we list the following: Theorem (2011). Suppose R is an indecomposable ring of char(R) = p and G is a group for which G0/Gp is finite. Then the following isomorphism is true: (*) U(RG) ∼= ∐ d/exp(G0/Gp) ∐ a(d) [U(R[ζd]) × [(G/ ∐ q 6=p Gq)Vp(R[ζd](G/ ∐ q6=p Gq))]] where a(d) = |{g∈G0/Gp:order(g)=d}| [R[ζd]:R] . In particular: (1) if G is p-splitting, then U(RG) ∼= ∐ d/exp(G0/Gp) ∐ a(d) [U(R[ζd]) × Vp(R[ζd](G/ ∐ q 6=p Gq))] × ∐ ∑ d/exp(G0/Gp) a(d) G/G0. (2) if Gp is a direct sum of cyclic groups, then U(RG) ∼= ∐ d/exp(G0/Gp) ∐ a(d) [U(R[ζd]) × (Vp(R[ζd](G/ ∐ q6=p Gq))/(G/ ∐ q 6=p Gq)p)]× × ∐ ∑ d/exp(G0/Gp) a(d) G/ ∐ q 6=p Gq. Moreover, the quotient Vp(R[ζd](G/ ∐ q 6=p Gq))/(G/ ∐ q 6=p Gq)p) is a direct sum of cyclic groups by [D] and can be characterized via the Ulm-Kaplansky invariants calculated in [Df]. Before stating and proving our chief attainment, we need two more preliminaries. Proposition 1. Let R = ∏ i∈I Ri be a direct product of subrings Ri where I is an index set, and F is a finite abelian group. Then the following isomorphism holds: RF ∼= ∏ i∈I RiF. 52 Peter Danchev CUBO 14, 1 (2012) Proof. It is straightforward and we leave it to the reader. 4 Lemma 2. Suppose G0/Gp is bounded. Then the following decomposition is true: G = M × B where M ∼= G/ ∐ q 6=p Gq is p-mixed and B ∼= ∐ q 6=p Gq ∼= G0/Gp is bounded. Proof. Since ∐ q 6=p Gq is bounded and is pure in G0 as its direct factor, whence pure in G, it follows that ∐ q 6=p Gq is a direct factor of G as well. Denoting B = ∐ q6=p Gq, one may write G = B × M where M ∼= G/B. It is obvious that M is p-mixed, i.e., M0 = Mp. 4 So, we come to our main achievement. Theorem 3. Let R be a ring of prime characteristic p which is a direct product of indecomposable rings Ri for some index set I, and let G be an abelian group such that G0/Gp is finite. Then the following isomorphism formula is fulfilled: (*) U(RG) ∼= [ ∐ i∈I U(Ri(G0/Gp))] × [Id(LM)Vp(LM)] for some commutative unitary ring L of prime characteristic p which is a direct product of inde- composable rings, and where, for all indices i ∈ I, U(Ri(G0/Gp)) ∼= ∐ d/exp(G0/Gp) ∐ ai(d) U(Ri[ζd]) with ai(d) = |{g∈G0/Gp:order(g)=d}| [Ri[ζd]:Ri] . In particular, the maximal divisible subgroup dU(RG) of U(RG) is completely described up to isomorphism. Proof. According to Lemma 2 one may write G = F × M where F ∼= ∐ q 6=p Gq is finite and M is p-mixed. Thus RG = (RF)M = LM where we put RF = L. Therefore, U(RG) = U(LM) = U(RF) × V(LM). Concerning V(LM) we may write V(LM) = Id(LM)Vp(LM) (see, e.g., [Dd] or [De]). On the other hand, owing to Proposition 1, L = RF = ( ∏ i∈I Ri)F ∼= ∏ i∈I RiF where each Ri is CUBO 14, 1 (2012) Units in Abelian Group Algebras Over Direct Products . . . 53 an indecomposable ring of characteristic p. Furthermore, since F is finite of exponent that inverts in R, and hence it inverts in each Ri, appealing to Theorem 4.4 and Remark 4.5 of [MN], every RiF is a finite direct sum of indecomposable subrings. Consequently, L is a commutative unitary ring of prime characteristic p which can be interpreted as a ring that is a direct product of indecomposable subrings. Moreover, U(RF) ∼= ∐ i∈I U(RiF), where U(RiF) has an explicit description for any index i. Thus formula (*) is deduced. Finally, observe that dU(RG) = dU(RF)×dV(LM) ∼= ∐ i∈I dU(RiF)×dV(LM). Since U(RiF), and hence dU(RiF), is already characterized above, and dV(LM) is classified in [Dd] and [De], we infer that the same can be said of dU(RG). 4 Remark. The proof of Theorem 2.7 from [MMN] contains a gap and so it is uncomplete. In fact, the authors claimed that they will assume that the splitting group is p-mixed. The reason is that the K-algebras isomorphism KG ∼= KH yields that K(G/ ∐ q 6=p Gq) ∼= K(H/ ∐ q6=p Hq) whenever K is a field of char(K) = p. But they need to show that G being splitting ensures that so is G/ ∐ q 6=p Gq. However, this was already done in [Db]. We close the work with the following problem. Conjecture. Suppose R is an indecomposable ring and G is a finite group of exponent which inverts in R. Then RG ∼= RH for some group H if, and only if, H is finite with the same exponent as that of G and RGp ∼= RHp for each prime number p. Notice that the sufficiency is trivial, because G and H being both bounded implies that G = ∐ p Gp and H = ∐ p Hp, whence RG ∼= ⊗RRGp and RH ∼= ⊗RRHp. Thus RGp ∼= RHp forces that RG ∼= RH, as desired. Received: October 2010. Revised: March 2011. References [D] P. V. Danchev, Commutative group algebras of σ-summable abelian groups, Proc. Amer. Math. Soc. (9) 125 (1997), 2559-2564. [Da] P. V. Danchev, Normed units in abelian group rings, Glasgow Math. J. (3) 43 (2001), 365-373. [Db] P. V. Danchev, Notes on the isomorphism and splitting problems for commutative modular group algebras, Cubo Math. J. (1) 9 (2007), 39-45. [Dc] P. V. Danchev, Warfield invariants in commutative group rings, J. Algebra Appl. (6) 8 (2009), 829- 836. 54 Peter Danchev CUBO 14, 1 (2012) [Dd] P. V. Danchev, Maximal divisible subgroups in p-mixed modular abelian group rings, Commun. Algebra (6) 39 (2011), 2210-2215. [De] P. V. Danchev, Maximal divisible subgroups in modular group rings of p-mixed abelian groups, Bull. Braz. Math. Soc. (1) 41 (2010), 63-72. [Df] P. V. Danchev, Ulm-Kaplansky invariants in commutative modular group rings, J. Algebra Number Theory Academia (2) 1 (2011), 127-134. [Dg] P. V. Danchev, Units in abelian group algebras over indecomposable rings, Studia “Babes Bolyai” - Mat. (4) 56 (2011), 3-6. [K] G. Karpilovsky, Units of commutative group algebras, Expo. Math. 8 (1990), 247-287. [M] W. L. May, Group algebras over finitely generated rings, J. Algebra 39 (1976), 483-511. [MMN] W. L. May, T. Zh. Mollov, N. A. Nachev, Isomorphism of modular group algebras of p-mixed abelian groups, Commun. Algebra 38 (2010), 1988-1999. [MN] T. Zh. Mollov and N. A. Nachev, Unit groups of commutative group rings, Commun. Algebra 34 (2006), 3835-3857.