Research Article

On dual-classical Krull dimension of rings

DOI: 10.2989/16073606.2022.2110960
Author(s): Sayed Malek Javdannezhad Shahid Rajaee Teacher Training University, Iran, Sayedeh Fatemeh Mousavinasab Shahid Chamran University of Ahvaz, Iran, Nasrin Shirali Shahid Chamran University of Ahvaz, Iran,

Abstract

An ideal L of a ring R is said to be strongly hollow (briefly, sh-ideal) if LA + B implies that either LA or LB, 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.

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