Commutative \(2\)-Euclidean rings (Q5954043)

A commutative ring \(R\) is called \(2\)-Euclidean relatively to the norm \(\mathcal N\) if for any elements \(a\in R{\backslash 0}\) and \(b\in R\) one of the following conditions holds: NEWLINENEWLINENEWLINE(a) there exist elements \(q, r \in R\) such that \(b=aq+r\) and \({\mathcal N}(r)<{\mathcal N}(a),\) NEWLINENEWLINENEWLINE(b) there exist elements \(q_1,q_2, r_1,r_2\in R\) such that \(b=aq_{1}+r_{1}\), \(a=r_{1}q_{2}+r_{2}\) and \({\mathcal N}(r_{2})<{\mathcal N}(a).\) NEWLINENEWLINENEWLINERecall that a commutative ring is called a Bezout ring if every of its finitely generated ideals is principal. The main result of the paper: NEWLINENEWLINENEWLINEA commutative \(2\)-Euclidean ring \(R\) is a ring with elementary reduction of matrices (i.e. every matrix with the entries from \(R\) is elementary equivalent to the diagonal matrix NEWLINE\[NEWLINE\text{diag}(\varepsilon_{1},\ldots,\varepsilon_{r},0,\ldots, 0),NEWLINE\]NEWLINE where \(\varepsilon _{i+1}R\subseteq \varepsilon _{i}R)\).