Adequate Properties of Matrix Divisors

Adequate rings and, in particular, adequate elements appear as a generalization of principal ideal rings in the investigation of the problem of matrix reduction. The main property of elements of these rings is the possibility of their representation in the form of the product of two factors satisfying certain conditions. We introduce the notion of adequacy in the case of noncommutative rings and study the properties of matrix divisors over the adequate rings. We also analyze the influence of changes in the Smith normal forms and transforming matrices on the adequate properties of matrix divisors.