Characterizations of almost PP-ring for three important classes of rings

Abstract

A ring is called an almost pp-ring if the annihilator of each element of R is generated by its idempotents.  We prove that for a ring R and an Abelian group G, if the group ring RG is an almost pp-ring then so is R.  Moreover, if G is a finite Abelian group then |G|⁻¹ ∈ R.  Then we give a counter example to the converse of this.  Also, we prove that RG is an almost pp-ring if and only if RH is an almost pp-ring for every subgroup H of G.  It is proved that the polynomial ring R[x] is an almost pp-ring if and only if R is an almost pp-ring.  Finally, we prove that the power series ring R[[x]] is an almost pp-ring if and only if for any two countable subsets S and T of R such that S ⊆ Ann_R(T), there exists an idempotent e ∈ Ann_R(T) such that b=be for all b ∈ S.