nLab
socle

Given a ring RR, the socle Soc(M)Soc(M) of a left RR-module MM is the (internal) sum of all simple submodules of MM. The correspondence MSoc(M)M\to Soc(M) is clearly a subfunctor of the identity functor RMod RMod{}_R Mod\to {}_R Mod. It is moreover left exact (but not a kernel functor in the sense of Goldman).

By the definition, the socle is a semisimple RR-module. If we assume the axiom of choice, than the socle of MM can be presented as a direct sum of some subfamily of all simple submodules of MM.

The notion of socle is important in representation theory.

Notice that the notion is dual to the notion of the radical Rad(M)Rad(M) which is the intersection of all maximal submodules of MM.

Revised on April 4, 2012 17:24:44 by Tim Porter (95.147.237.221)