Standard Forms of the Matrices Over Rings with Respect to Various Types of Equivalence and Their Applications to the Theory of Matrix Factorization and Matrix Equations

We present a survey of the results investigations one fields dealing with equivalence matrices originated by P. S. Kazimirs’kii and then continued developed his colleagues. formulate standard forms polynomial their finite sets respect to semiscalar generalized pairs over rings. Their applications development methods matrix factorization, solving equations, description structure solutions these equations and, in particular, Sylvester linear Diophantine other problems are indicated.