On adequacy of full matrices

Abstract

This paper deals with the following question:whether a ring of matrices or classes of matrices over an adequate ring or elementary divisor ring inherits the property of adequacy? The property to being adequate in matrix rings over adequate and commutative elementary divisor rings is studied.Let us denote by $\mathfrak{A}$ and $\mathfrak{E}$ an adequate and elementary divisor domains, respectively. Also $\mathfrak{A}_2$ and $\mathfrak{E}_2$ denote a rings of $2 \times 2$ matrices over them. We prove that full nonsingular matrices from $\mathfrak{A}_2$ are adequate in $\mathfrak{A}_2$ and full singular matrices from $\mathfrak{E}_2$ are adequate in the set of full matrices in $\mathfrak{E}_2$.