Solution
= Solution
An $A$-<module> is \b[completely reducible] if it is a <direct sum of modules> each of which is a <simple module>:
$$
V=\bigoplus_{j\in J}S_j.
$$
Here a <simple module> is nonzero and has no <submodules> other than zero and itself. This is precisely the definition of a <semisimple module>. The indexing set need not be finite unless $V$ has finite <dimension>; the zero <module> is included as the empty <direct sum of modules>.