A characterization of commutative rings whose maximal ideal spectrum is Noetherian

Abstract

An ideal [Formula: see text] of a ring [Formula: see text] is called pseudo-irreducible if [Formula: see text] cannot be written as an intersection of two comaximal proper ideals of [Formula: see text]. In this paper, it is shown that the maximal spectrum of [Formula: see text] is Noetherian if and only if every proper ideal of [Formula: see text] can be expressed as a finite intersection of pseudo-irreducible ideals. Using a result of Hochster, we characterize all [Formula: see text] quasi-compact Noetherian topological spaces.

J-Noetherian Bezout domain which is not a ring of stable range 1

“…Since R is not a ring of stable range 1, then in R there exist a nonunit neat element a. Since R is a complete factorization ring [2], then a has a factorization a = a 1 . .…”