Completely-reducible module

2020 Mathematics Subject Classification: Primary: 13C [MSN][ZBL]

semi-simple module

A module $M$ over an associative ring $R$ which can be represented as the sum of its irreducible $R$-submodules (cf. Irreducible module). Equivalent definitions are: 1) $M$ is the sum of its minimal submodules; 2) $M$ is isomorphic to a direct sum of irreducible modules; or 3) $M$ coincides with its 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. 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

[1]	J. Lambek, "Lectures on rings and modules" , Blaisdell (1966)
[2]	N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)