 While the direct sum is uniquely determined by , the isomorphism giving the decomposition itself is "not unique" in general.For instance if is actually a field, then all occurring ideals must be zero, and one obtains the decomposition of a finite dimensional vector space into a direct sum of one-dimensional subspaces; the number of such factors is fixed, namely the dimension of the space, but there is in a lot of freedom for choosing the subspaces themselves (if ).The nonzero formula_3 elements, together with the number of formula_3 which are zero, form a complete set of invariants for the module.