Rings with elementary reduction of matrices (Q2740357)

A ring \(R\) is said to have elementary reduction if every matrix \(A\) over \(R\) is elementary equivalent to the canonical diagonal matrix \(\text{diag}(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_r,0,\dots,0)\) where \(\varepsilon_iR\cap R\varepsilon_i\supseteq R\varepsilon_{i+1}R\), \(i=1,2,\dots,r-1\). In the paper, necessary and sufficient conditions for a quasi-Euclidean ring are obtained to be a ring with elementary reduction of matrices. It is also shown that a quasi-Euclidean ring \(R\) every non-invertible element of which is contained in an at most countable set of maximal ideals is a ring with elementary reduction of matrices. The same conclusions are made under some restrictions for semilocal Bézout rings and 2-Euclidean domains.