We de_ne the canonical presentation of an R-n-module, in terms of its largest n-submodule with zero and of an idempotent commutative n-group. We give a construction for the free R-n-module with zero, as well as a canonical presentation for the free R-n-module. We give the number of zero-idempotents of a _nitely generated free R-n-module. The last theorem states that, for n > 3, free R-nmodules are isomorphic if and only if their free generating sets have the same cardinality.