A Bezout ring of stable range 2 which has square stable range 1

Abstract

In this article, we introduced the concept of a ring of stable range 2 which has square stable range 1. We proved that a Hermitian ring R which has (right) square stable range 1 is an elementary divisor ring if and only if R is a duo ring of neat range 1. And we found that a commutative Hermitian ring R is a Toeplitz ring if and only if R is a ring of (right) square range 1. We proved that if R be a commutative elementary divisor ring of (right) square stable range 1, then for any matrix one can find invertible Toeplitz matrices P and Q such that where e1 is a divisor of e2.