On Generalized Stable Ideals

Abstract

An ideal I of a ring R is generalized stable in case aR + bR = R with a ∈ I b ∈ R implies that there exist s t ∈ 1 + I such that s a + by t = 1 for a y ∈ R. We establish, in this article, necessary and sufficient conditions for an ideal of a regular ring to be generalized stable. It is shown that every regular square matrix over such ideals admits a diagonal reduction. These extend the corresponding results of generalized stable regular rings.

Local Comparability of Exchange Ideals

“…In [2], Chen proved that comparability of modules over one-sided unit-regular rings is Morita invariant, in terms of comparability. In [3], the author considered a class of ideals in a regular ring. In [4], the author introduced and investigated a kind of quasi-stable exchange ideals.…”