nLab
socle

Given a ring $R$, the socle $\mathrm{Soc}(M)$ of a left $R$-module $M$ is the (internal) sum of all simple submodules of $M$. The correspondence $M\to \mathrm{Soc}(M)$ is clearly a subfunctor of the identity functor ${}_R\mathrm{Mod}\to {}_R\mathrm{Mod}$. 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, than 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 $\mathrm{Rad}(M)$ which is the intersection of all maximal submodules of $M$.

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