Proposition 5

(i) Let $(E_i)_{i \in I}, (F_i)_{i \in I}$ be two families of A-modules with the same indexing set I; for every family of linear mappings $f_i : E_i \to F_i$ ($i \in I$), the mapping $f : (x_i) \to (f_i(x_i))$ of $\prod_i E_i$ into $\prod_i F_i$ (sometimes denoted by $\prod_i f_i$) is linear.

(ii) Let $(G_i)_{i \in I}$ be a third family of A-modules with I as indexing set and, for all $i \in I,$ let $g_i : F_i \to G_i$ be a linear mapping; let $g = \prod_i g_i$. For each of the sequences $E_i \xrightarrow{f_i} F_i \xrightarrow{g_i} G_i$ to be exact, it is necessary and sufficient that the sequence

be exact.

Assertion (i) follows immediately from the definitions. On the other hand, to say that $y = (y_i)$ belongs to $\operatorname{Ker}(g)$ means that $g_i(y_i) = 0$ for all $i \in I$ and hence that $y_i \in \operatorname{Ker}(g_i)$ for all $i \in I$; similarly, to say that $y$ belongs to $\operatorname{Im}(f)$ means that there exists $x = (x_i) \in \prod_i E_i$ such that $y = f(x)$, which is equivalent to saying that $y_i = f_i(x_i)$ for all $i \in I$ or also that $y_i \in \operatorname{Im}(f_i)$ for all $i \in I$; whence (ii).