GPF-Properties of Group Rings

Abstract

All rings R in this article are assumed to be commutative with unity 1 ≠ 0. A ring R is called a GPF−ring if for every a ∈ R there exists a positive integer n such that the annihilator ideal Ann_R (aⁿ) is pure. We prove that for a ring R and an Abelian group G, if the group ring RG is a GPF−ring then so is R. Moreover, if G is a finite Abelian group then |G| is a unit or a zero-divisor in R. We prove that if G is a group such that for every nontrivial subgroup H of G, [G : H] < ∞, then the group ring RG is a GPF−ring if and only if RH is a GPF−ring for each finitely generated subgroup H of G. It is proved that if R is a local ring and RG is a U−group ring, then RG is a GPF−ring if and only if R is a GPF−ring and p ∈ Nil(R). Finally, we prove that if R is a semisimple ring and G is a finite group such that |G|⁻¹ ∈ R, then RG is a GPF−ring if and only if RG is a PF−ring.

Key words and phrases. GPF−ring, group ring, local ring, U−group ring.

1991 Mathematics Subject Classification. 16D40, 16S34, 16S85