Generalized equivalence of pairs of matrices

Pairs (A₁B₁) and (A₂B₂) of matrices over a principal ideal domain R are called the generalized equivalent pairs if A₂=UA₁V₁B₂=UB₁V₂ for some invertible matrices UV₁V₂ over R. A special form is established to which a pair of matrices can be reduced by means of generalized equivalent transformations. Besides necessary and sufficient conditions are found, under which a pair of matrices is generalized equivalent to a pair of diagonal matrices. Applications are made to study the divisibility of matrices and multiplicative property of the Smith normal form.