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On the euclideanity of matrix modules over a given Euclidean ring. (Russian) Zbl 0586.16003

Let \(R\) be a ring with 1 and \(R_{mn}\) be the set of \((m\times n)\)-matrices over \(R\). A subset \(\Sigma\) of a right module \(M\) is said to be Euclidean, if there exists a map \(\Phi\) from \(\Sigma\) into a completely ordered set \(W\) such that \[ \forall a\in M\quad \forall b\in \Sigma \setminus \{0\}\quad \exists q\in R\quad \exists r\in \Sigma \quad ((a=bq+r)\;\&\;(\Phi (r)<\Phi (b)). \] The map \(\Phi\) is called a \(\Sigma\)-Euclidean algorithm of \(M\) and \(\Phi\) \((\Sigma)\) is by definition the type of \(\Phi\). A module \(M\) is called Euclidean, if \(M\) is a Euclidean subset. If \(R\) is a Euclidean right \(R\)-module and \(m\leq n\), then \(R_{mn}\) is a Euclidean right \(R_{nn}\)-module. If \(R\) is an integral domain and \(m\leq n\), then \(\{A\mid A\in R_{mn},\;\text{rank}\,A\geq r\}\) \((0\leq r\leq m)\) are Euclidean subsets of the right \(R_{nn}\)-module \(R_{mn}\). Then if \(R\) is no skew-field and \(n>1\), then \(R_{nn}\) has no \(R_{nn}\)-Euclidean algorithm of type \(\leq \omega\).

MSC:

16U30 Divisibility, noncommutative UFDs
16S50 Endomorphism rings; matrix rings
16U10 Integral domains (associative rings and algebras)
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