On simple factorization of invertible matrices

Abstract

Let I be an ideal of a ring R. We say that R is a generalized I-stable ring provided that aR+bR=R with a ∈ 1+I,b ∈ R implies that there exists a y ∈ R such that a+by ∈ K(R), where K(R)={x ∈ R ∣ ∃ s, t ∈ R such that sxt=1}. Let R be a generalized I-stable ring. Then every A ∈ GL_n (I) is the product of 13n−12 simple matrices. Furthermore, we prove that A is the product of n simple matrices if I has stable rank one. This generalizes the results of Vaserstein and Wheland on rings having stable rank one.

Keywords:

• Invertible matrices
• Generalized ideal-stable rings
• Stable rank one

Keywords:

• 2000 Mathematics Subject Classifications: 15A23
• 15A33
• 19B10

Acknowledgement

It is a pleasure to thank the referee for the suggestions which lead to the new version.