GPF-Properties of Group Rings
Authors : Huda Odetallah, Hasan Al-Ezeh and Emad Abu Osba
Abstract :All rings R in this article are assumed to be commutative with unity 1 6= 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 AnnR (a n) 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
Keywords : GPF−ring, group ring, local ring, U−group ring
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