Arithmetic of matrices over rings

TL;DR

This work explores the arithmetic and factorization properties of matrices over specific classes of rings, focusing on matrix factorizations, subgroup structures, and matrix forms over rings with stable range 1.5.

Contribution

It introduces new insights into matrix factorization theory over rings of stable range 1.5 and links these properties to subgroup structures and matrix normal forms.

Findings

Matrix factorization theory over rings of stable range 1.5 developed

Relationship established between matrix factorizations and subgroup properties

Properties of matrices over certain principal ideal domains analyzed

Abstract

The book is devoted to investigation of arithmetic of the matrix rings over certain classes of commutative finitely generated principal ideals domains. We mainly concentrate on constructing of the matrix factorization theory. We reveal a close relationship between the matrix factorization and specific properties of subgroups of the complete linear group and the special normal form of matrices with respect to unilateral equivalence. The properties of matrices over rings of stable range 1.5 are thoroughly studied. The book is intended for experts in the ring theory and linear algebra, senior and post-graduate students.