nLab socle

Definition

Given a ring $R$, the socle $Soc(M)$ of a left $R$-module $M$ is the internal sum of all simple submodules of $M$.

Properties

Given a ring $R$, the correspondence $M\mapsto Soc(M)$ is clearly a subfunctor of the identity functor ${}_R Mod\to {}_R Mod$. It is moreover left exact (but not a kernel functor in the sense of Goldman in general).

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

The notion of socle is important in representation theory.

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

Literature

• Louis H. Rowen, Ring theory – Student edition, Academic Press (1991, 2012)
• Joachim Lambek, Lectures on rings and modules, Waltham Mass. 1966
• wikipedia socle (mathematics)