Unit groups of commutative group rings

Abstract

Let R be a commutative ring with identity. An algebraic element over R is called an “integral algebraic element” over R if there exists a minimal polynomial of over R which is monic. Let R be a direct product of m indecomposable rings Ri m ∈ . Denote by RG the group ring of G over R and by R∗ the multiplicative group of R. Let G be a finite Abelian group of exponent n and n ∈ R. In this paper we give a decomposition of RG, up to isomorphism, into a direct sum of extensions of the ring R, taking into account the number of the repetitions of these extensions. If the ring R is a field, then this result is proved in Perlis and Walker (1950). Let G be a splitting Abelian group with a torsion subgroup G0. Denote by Gp the p-component of G. We give a description of the unit group URG of RG in the following cases: (i) when Ri is a ring of prime characteristic pi, G0/Gpi is finite and the exponent of G0/Gpi belongs to R∗ i ; (ii) when Ri is of characteristic zero, Ri has no nilpotents, G0 is finite of exponent n and n ∈ R∗ i . For the establishment of these results we prove that if the ring R is indecomposable and n ∈ R∗, then: (i) the cyclotomic polynomial nx has a unique decomposition in a product of monic irreducible factors over R; and (ii) if and are integral algebraic elements over R which are roots of monic irreducible divisors of the cyclotomic polynomial nx , then the rings R and R are isomorphic.