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.

Key Words:

• Diagonal reduction
• Generalized stable ideal
• Regular rings

2000 Mathematics Subject Classification:

• 16E50
• 16B10

ACKNOWLEDGMENTS

The author would like to thank the referee for his/her many corrections and suggestions, which considerably improved article and lead to the new version.

Notes

Communicated by J. L. Gomez Pardo.