d-ideals, fd-ideals and prime ideals

Abstract

Let R be a commutative ring. An ideal I of R is called a d-ideal (fd- ideal) provided that for each a ∈ I (finite subset F of I) and b ∈ R, Ann(a) ⊆ Ann(b) (Ann(F) ⊆ Ann(b)) implies that b ∈ I. It is shown that, the class of z⁰-ideals (hence all sz⁰-ideals), maximal ideals in an Artinian or in a Kasch ring, annihilator ideals, and minimal prime ideals over a d-ideal are some distinguished classes of d-ideals. Furthermore, we introduce the class of fd-ideals as a subclass of d-ideals in a commutative ring R. In this regard, it is proved that the ring R is a classical ring with property (A) if and only if every maximal ideal of R is an fd-ideal. The necessary and sufficient condition for which every prime fd-ideal of a ring R being a maximal or a minimal prime ideal is given. Moreover, the rings for which their prime d-ideals are z⁰-ideals are characterized. Finally, we prove that every prime fd-ideal of a ring R is a minimal prime ideal if and only if for each a ∈ R there exists a finitely generated ideal I ⊆ √Ann(an), for some n ∈ N such that Ann(a, I) = 0. As a consequence, every prime fd-ideal in a reduced ring R is a minimal prime ideal if and only if X = Min(R) is a compact space.

Mathematics Subject Classification (2010): Primary 13A15; Secondary 54C40.

Keywords: d-ideal, fd-ideal, property (A), classical ring, z-ideal, z⁰-ideal, Kasch ring, compact space