View source for Completely-reducible module

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A module $A$ over an associative ring $R$ which can be represented as the sum of its irreducible $R$-submodules (cf. [[Irreducible module|Irreducible module]]). Equivalent definitions are: 1) $A$ is the sum of its minimal submodules; 2) $A$ is isomorphic to a direct sum of irreducible modules; or 3) $A$ coincides with its [[Socle|socle]]. A submodule and a quotient module of a completely-reducible module are also completely reducible. The lattice of submodules of a module $M$ is a lattice with complements if and only if $M$ is completely reducible.

If all right $R$-modules over a ring $R$ are completely reducible, all left $R$-modules are completely reducible as well, and vice versa; $R$ is then said to be a completely-reducible ring or a [[Classical semi-simple ring|classical semi-simple ring]]. For a ring $R$ to be completely reducible it is sufficient for it to be completely reducible when regarded as a left (right) module over itself.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Lambek,   "Lectures on rings and modules" , Blaisdell  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Jacobson,   "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR></table>