Proposition 12

Let E be an A-module.

(i) If E is the direct sum of a family $(M_\lambda)_{\lambda \in L}$ of submodules and, for all $x \in E$, $e_\lambda(x)$ is the component of x in $M_\lambda$, $(e_\lambda)$ is an orthogonal family of projectors such that $x = \sum_{\lambda \in L} e_\lambda(x)$ for all $x \in E$.

(ii) Conversely, if $(e_\lambda)_{\lambda \in L}$ is an orthogonal family of projectors in $\operatorname{End}(E)$ such that $x = \sum_{\lambda \in L} e_\lambda(x)$ for all $x \in E$, E is the direct sum of the family of submodules $M_\lambda = e_\lambda(E)$.

Property (i) follows from the definitions and (ii) is a special case of no. 6, Corollary 2 to Proposition 6, applied to the canonical injections $M_\lambda \to E$ and the mappings $e_\lambda : E \to M_\lambda$.

Note that when L is finite the condition $x = \sum_{\lambda \in L} e_\lambda(x)$ for all $x \in E$ may also be written in $\operatorname{End}(E)$.

(31)