Factorization of matrices over elementary divisor domain

We propose constructive criteria of divisibility and associativity of matrices over commutative elementary divisor ring without zero divisors. On this base, the explicit form for all non-associated divisors which have prescribed canonical diagonal forms (c.d.f.) is indicated. A relation between c.d.f. for matrix and c.d.f. for its divisors is established. The uniqueness theorem is proved.