Corollary 1

Under the hypotheses of Proposition 23, suppose that each M_i is monogenous and that E is the direct sum of a second family (N_\lambda)_{\lambda \in L} of non-zero monogenous submodules. Then Card(L) = Card(I).

If b_\lambda is a generator of N_\lambda, the set of b_\lambda is a generating system of E, hence Card(L) \geq Card(I). In particular L is infinite and, exchanging the roles of (M_i) and (N_\lambda), similarly Card(I) \geq Card(L), whence the corollary.