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.).
2000 Mathematics Subject Classification:
Acknowledgements
The authors are grateful to the referee for suggestions that helped improve the presentation of the paper.