# 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)
How to Cite This Entry:
Completely-reducible module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completely-reducible_module&oldid=36913
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article