Proposition 10

Let M be a semisimple A-module. Let $(M_i)_{i\in I}$ be a family of simple submodules with direct sum M. The following properties are equivalent:

(i) M has finite length.

(ii) M is Artinian.

(iii) M is Noetherian.

(iv) M is finitely generated.

(v) I is finite.

If M has these properties, then the length of M is equal to the cardinal of I.

If the set I is finite, then M has properties (i), (ii), (iii), and (iv). Suppose that the set I is infinite. By Example 2 of VIII, p. 2, the module M is neither Artinian nor Noetherian; because every module of finite length is Artinian and Noetherian (VIII, p. 2, Proposition 1), M also does not have finite length. Finally, every element of M belongs to the sum of a finite number of submodules $M_i$, so M is not finitely generated. This proves the equivalence of properties (i) through (v). If these hold, then we have long(M) $=\sum_{i\in I}$ long(M$_i$) $=$ Card(I) (II, §1, No. 10, p. 213, Corollary 5).