Proposition 6

Let $(E_i)_{i \in I}$ be a family of A-modules, $M$ an A-module and, for all $i \in I$, let $f_i : E_i \to M$ be a linear mapping. Then there exists one and only one linear mapping $g : \bigoplus_{i \in I} E_i \to M$ such that, for all $i \in I$:

(21)

By virtue of (20), if $g$ exists, then necessarily, for all $x \in \bigoplus_{i \in I} E_i$,

whence the uniqueness of $g$. Conversely, setting $g(x) = \sum_i f_i(\mathrm{pr}_i(x))$ for all $x \in \bigoplus_{i \in I} E_i$, it is immediately verified that a linear mapping has been defined satisfying the conditions of the statement.

When no confusion can arise, we write $g = \sum_{i \in I} f_i$ (which is contrary to the conventions of I, § 2, no. 1, when the family $(f_i)$ is not of finite support).

In particular, if J is any subset of I, the canonical injections $j_i$ for $i \in J$ define a canonical linear mapping $j_J : \bigoplus_{i \in J} E_i \to \bigoplus_{i \in I} E_i$, which associates with each $(x_i)_{i \in I}$ the element $(x'_i)_{i \in I}$ such that $x'_i = x_i$ for $i \in J$, $x'_i = 0$ for $i \notin J$; this mapping is obviously injective. Moreover, if $(J_\lambda)_{\lambda \in L}$ is a partition of I, the mapping $i : \bigoplus_{\lambda \in L} \left( \bigoplus_{i \in J_\lambda} E_i \right) \to \bigoplus_{i \in I} E_i$ corresponding to the family $(j_{J_\lambda})$ by Proposition 6 is an isomorphism called canonical ("associativity" of direct sums).