Unit Groups of Commutative Group Rings
Todor Zh. Mollov, Nako A. Nachev · Communications in Algebra · 2006
Let R be a commutative ring with identity. An algebraic element α over R is called an “integral algebraic element” over R if there exists a minimal polynomial of α over R which is monic. Let R be a direct product of m indecomposable rings R i , m ∈ ℕ. 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 Perlis , S. , Walker , G. L. ( 1950 ). Abelian group algebras of finite order . Trans. Amer. Math. Soc. 68 : 420 – 426 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). Let G be a splitting Abelian group with a torsion subgroup G 0. Denote by G p the p -component of G. We give a description of the unit group U(RG) of RG in the following cases: i. when R i is a ring of prime characteristic p i , G 0/G p i is finite and the exponent of G 0/G p i belongs to R i *;ii. when R i is of characteristic zero, R i has no nilpotents, G 0 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 Φ n (x) has a unique decomposition in a product of monic irreducible factors over R; andii. if α and β are integral algebraic elements over R which are roots of monic irreducible divisors of the cyclotomic polynomial Φ n (x), then the rings R[α] and R[β] are isomorphic.