Proposition 11

Let $(M_\iota)_{\iota \in I}$ be a family of submodules of an A-module $E$. The following properties are equivalent:

(a) The submodule $\sum_{\iota \in I} M_\iota$ is the direct sum of the family $(M_\iota)_{\iota \in I}$.

(b) The relation $\sum_{i \in I} x_i = 0$, where $x_i \in M_i$ for all $i \in I$, implies $x_i = 0$ for all $i \in I$.

(c) For all $\kappa \in I$, the intersection of $M_\kappa$ and $\sum_{i \neq \kappa} M_i$ reduces to 0.

It is immediate that (a) and (b) are equivalent, since the relation $\sum_i x_i = \sum_i y_i$ is equivalent to $\sum_i (x_i - y_i) = 0$. On the other hand, by virtue of Definition 6, (a) implies (c) by the uniqueness of the expression of an element of $\bigoplus_{i \in I} M_i$ as a direct sum of elements $x_i \in M_i$. Finally, the relation $\sum_i x_i = 0$, where $x_i \in M_i$ for all $i$, can be written, for all $\kappa \in I$, $x_\kappa = \sum_{i \neq \kappa} (-x_i)$; condition (c) then implies $x_\kappa = 0$ for all $\kappa \in I$, hence (c) implies (b).