Given a ring , the socle of a left -module is the (internal) sum of all simple submodules of . The correspondence is clearly a subfunctor of the identity functor . It is moreover left exact (but not a kernel functor in the sense of Goldman).
By the definition, the socle is a semisimple -module. If we assume the axiom of choice, then the socle of can be presented as a direct sum of some subfamily of all simple submodules of .
The notion of socle is important in representation theory.
Notice that the notion is dual to the notion of the radical which is the intersection of all maximal submodules of .