Proposition 7

(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 restriction to $\bigoplus_{i \in I} E_i$ of the linear mapping $(x_i) \mapsto (f_i(x_i))$ is a linear mapping $f : \bigoplus_{i \in I} E_i \to \bigoplus_{i \in I} F_i$ denoted by $\bigoplus_{i \in I} f_i$ or $\bigoplus_i f_i$ (where $f = f_p \oplus f_{p+1} \oplus \cdots \oplus f_q$ if $I = [p, q]$ is an interval in $\mathbf{Z}$*).*

(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; we write $g = \bigoplus_i g_i$. 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.

Obviously, for all $(x_i) \in \bigoplus_i E_i$, the family $(f_i(x_i))$ has finite support, whence (i). On the other hand, to say that an element $y = (y_i)$ of $\bigoplus_i F_i$ belongs to $\mathrm{Ker}(g)$ means that $y_i \in \mathrm{Ker}(g_i)$ for all $i \in I$ (no. 5, Proposition 5); similarly, if $y_i \in \mathrm{Im}(f_i)$ for all $i \in I$, there exists for each $i \in I$ an $x_i \in E_i$ such that $y_i = f_i(x_i)$ and when $y_i = 0$, it may be supposed that $x_i = 0$; hence $y \in \mathrm{Im}(f)$ and the converse is obvious.