On the lattice of annihilator ideals and its applications

Abstract

In this paper we study the properties of (the set of right annihilator ideals of R) as a lattice. We prove that is a Boolean algebra if and only if R is a semiprime ring. For a quasi Armandariz ring R, we show that the lattices and are isomorphic. In general, for any ring R, we show that the lattices and are isomorphic. We answer two questions related to property (a.c.) that were raised by Hong, Kim, Lee and Nielsen. To do this, we introduce a new class of rings. We say R is right strongly quasi Armandariz (-Armandariz) if for each there exists such that The class of right -Armandariz rings is between right -Baer rings and Armandariz rings. For a right -Armandariz ring R, it is proved that R has the right annihilator condition (a.c.) if and only if R[x] has right the annihilator condition (a.c.). We conclude that if R is a quasi Armandariz ring and has right (a.c.), then R[x] has right (a.c.).