The module is determined up to isomorphism by specifying its free rank , and for class of associated irreducible elements  and each positive integer  the number of times that  occurs among the elementary divisors. The elementary divisors can be obtained from the list of invariant factors of the module by decomposing each of them as far as possible into pairwise relatively prime (non-unit) factors, which will be powers of irreducible elements.
The module's structure is determined by its free rank and the frequency of certain irreducible elements within its elementary divisors. These elementary divisors are derived from the list of invariant factors of the module by breaking them down into pairwise relatively prime (non-unit) factors, which are powers of irreducible elements.