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.
