Proposition 1

Let M be an A-module. The following properties are equivalent:

(i) There exists a set I such that M is isomorphic to $S^{(I)}$.

(ii) The module M is the direct sum of a family of submodules isomorphic to S.

(iii) The module M is the sum of a family of submodules isomorphic to S.

(iv) There exists a left vector space V over the field D such that the A-module M is isomorphic to $S\otimes_DV$.

The equivalence of (i) and (ii) is immediate, and that of (ii) and (iii) follows from Theorem 1 of VIII, p. 56, applied to the case N = 0. Every left vector space over D is isomorphic to a vector space of the form $D_s^{(I)}$, where I is a set (II, §7, No. 1, p. 292, Theorem 1). Since the tensor product commutes with direct sums, (i) is equivalent to (iv).