Stability in Commutative Rings

Abstract

Let R be a commutative ring with zero-divisors and I an ideal of R . I is said to be ES-stable if JI = I 2 for some invertible ideal J ⊆ I , and I is said to be a weakly ES-stable ideal if there is an invertible fractional ideal J and an idempotent fractional ideal E of R such that I = JE . We prove useful facts for weakly ES-stability and investigate this stability in Noetherian-like settings. Moreover, we discuss a question of A. Mimouni on locally weakly ES-stable rings: is a locally weakly ES-stable domain of finite character weakly ES-stable?

ES-w-stability

“…An ideal I of an integral domain R is said to be a weakly ES-stable ideal if there is an invertible fractional ideal J and an idempotent fractional ideal E of R such that I ¼ JE. Recentley, the concepts of SV-stability, ES-stability and weakly ES-stability are extended to commutative rings with zerodivisors in [7,8].…”

“…The purpose of this paper is to study w-operation analogue of some facts that have been proven for ES-stable and weakly ES-stable domains in [8,47]. A nonzero ideal I of an integral domain R is called weakly ES-w-stable if I w ¼ ðJEÞ w for some t-invertible fractional ideal J of R and w-idempotent fractional ideal E of R. We define R to be a weakly ES-w-stable domain if every nonzero ideal of R is weakly ES-w-stable.…”