@article{Dzhaliuk_Petrychkovych_2021, title={Equivalence of matrices in the ring $M(n, R)$ and its subrings}, volume={73}, url={http://umj.imath.kiev.ua/index.php/umj/article/view/6858}, DOI={10.37863/umzh.v73i12.6858}, abstractNote={<p>UDC 512.64+512.55</p> <p>In this article, we consider the equivalence of matrices in the ring $M(n, R)$ and in its subrings of block triangular matrices $M<span style="font-size: 11.6667px;">_{BT}</span> (n_1, . . . , n_k, R)$ and block diagonal matrices $M<span style="font-size: 11.6667px;">_{BD}</span> (n_1, . . . , n_k, R)$ where $R$ is a commutative principal ideal domain, and investigate the connections between these equivalences. Under the conditions that the block triangular matrices are block diagonalizable, i.e., equivalent to their main block diagonals, we establish that these matrices are equivalent in the ring $M<span style="font-size: 11.6667px;">_{BT}</span> (n_1, . . . , n_k, R)$ of block triangular matrices if and only if their main diagonals are equivalent in the subring $M<span style="font-size: 11.6667px;">_{BD}</span> (n_1, . . . , n_k, R)$ of block diagonal matrices, i.e., the corresponding diagonal blocks of these matrices are equivalent. We also prove that if block triangular matrices $A$ and $B$ with the Smith normal forms $S(A) = S(B)$ are equivalent to the Smith normal forms in the subring $M<span style="font-size: 11.6667px;">_{BT}</span> (n_1, . . . , n_k, R)$, then these matrices are equivalent in the subring $M<span style="font-size: 11.6667px;">_{BT}</span> (n_1, . . . , n_k, R)$.</p>}, number={12}, journal={Ukrainsâ€™kyi Matematychnyi Zhurnal}, author={Dzhaliuk , N. S. and Petrychkovych , V. M.}, year={2021}, month={Dec.}, pages={1612 - 1618} }