On dual-classical Krull dimension of rings

Abstract

An ideal L of a ring R is said to be strongly hollow (briefly, sh-ideal) if L ⊆ A+B implies that either L ⊆ A or L ⊆ B, for every pair A and B of  ideals of R. We first, study and investigate these ideals. Even though, sh-ideals are apparently dual of the strongly irreducible ideals, but  as we shall see, they almost always behave as dual of prime ideals. In this paper we dualize some of the basic results of prime ideals for  sh-ideals. Using sh-ideals, we also give a new notion for dual-classical Krull dimension of rings that is actually good behavior. Then we  dualize almost all important results on classical Krull dimension. In particular, we prove that if R satisfies DCC on sh-ideals, then it has  dual-classical Krull dimension and in case R is a distributive ring (D-ring, for short), then the converse is true.