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, then 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.

Last revised on October 21, 2016 at 02:56:37. See the history of this page for a list of all contributions to it.