Definition 6

An A-module $E$ is said to be the direct sum of a family $(M_\iota)_{\iota \in I}$ of submodules of $E$ if the canonical mapping $\bigoplus_{\iota \in I} M_\iota \to E$ (no. 6) is an isomorphism.

It amounts to the same to say that every $x \in E$ can be written in a unique way in the form $x = \sum_{\iota \in I} x_\iota$, where $x_\iota \in E_\iota$ for all $\iota \in I$; the element $x_\iota$ thus corresponding to $x$ is called the component of $x$ in $E_\iota$; the mapping $x \mapsto x_\iota$ is linear.