socle

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

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 $M$.

- Joachim Lambek,
*Lectures on rings and modules*, Waltham Mass. 1966 - wikipedia socle (mathematics)