Reduction of invertible matrices by two-sided transformations from Zelisko groups to a simpler form

Abstract

Zelisko group originated in the study of invertible matrices that reduce a matrix to the Smith form. In the calculations related to finding the Smith form of matrix product, their greatest common divisor and least common multiple, the problem of reducing an invertible matrices by two-sided transformations from Zelisko groups to a simpler form arises. In the article, exactly such a form was obtained. We also establish the relationship between the stable range of a ring and the representation of invertible matrices as a product of three factors, two of which belong to Zelisko groups.