Flat ideals and stability in integral domains

Abstract

We introduce the concept of quasi-stable ideal in an integral\ud domain D (a nonzero fractional ideal I of D is quasi-stable if it\ud is flat in its endomorphism ring (I : I)) and study properties of\ud domains in which each nonzero fractional ideal is quasi-stable.\ud We investigate some questions about flatness that were raised by\ud S. Glaz and W.V. Vasconcelos in their 1977 paper [17]

Integral Domains in Which Every Nonzero w-Flat Ideal Is w-Invertible

“…It is well known that an ideal I of D is flat if and only if ID P is flat for all prime ideals P of D [10] (Proposition 3.10); so a nonzero flat ideal is w-flat. Additionally, it was shown that a nonzero flat ideal is a t-ideal [11] (Theorem 1.4). Hence we have the following lemma.…”

On LPI rings

“…In 2010, Picozza and Tartarone introduced the notion of quasi-stable ideals as ideals I that are flat in their ring homomorphisms (I : I). They also studied domains in which every ideal is quasi-stable, and proved that Glaz-Vascocncelos conjecture is false ( [20]).…”

STAR-INVERTIBILITY AND t-FINITE CHARACTER IN INTEGRAL DOMAINS

“…In this paper, we further generalize these results. As pointed out in [13], the finite character on maximal ideal is not necessary to have that locally principal ideals are invertible. In fact, Noetherian domains have obviously this property, but they have not necessarily the finite character.…”

“…In fact, Noetherian domains have obviously this property, but they have not necessarily the finite character. However, Noetherian domains have the t-finite character, which turns out to be a sufficient condition to obtain the invertibility of locally principal ideals (on the other hand, Example 2.3 shows that t-finite character is not necessary, answering a question posed in [13]).…”