Proposition 23

Let E be an A-module, the direct sum of an infinite family (M_i)_{i \in I} of non-zero submodules. For every generating system S of E, Card(S) \geq Card(I).

For all x \in S, let C_x be the finite set of indices i \in I such that the component of x in M_i is \neq 0 and let C = \bigcup_{x \in S} C_x. Every x \in S belongs by definition to the submodule of E the direct sum of the M_i for i \in C and the hypothesis that S generates E implies therefore that C = I; as I is by hypothesis infinite, so is S (Set Theory, III, § 5, no. 1, Corollary 1 to Proposition 1); therefore Card(I) = Card(C) \leq Card(S) (Set Theory, III, § 6, no. 3, Corollary 3 to Theorem 2).