# Radical of a module

In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle soc(M) of M.

## Definition

Let R be a ring and M a left R-module. A submodule N of M is called maximal or cosimple if the quotient M/N is a simple module. The radical of the module M is the intersection of all maximal submodules of M,

${\displaystyle {\mathrm {rad} }(M)=\bigcap \{N\mid N{\mbox{ is a maximal submodule of M}}\}\,}$

Equivalently,

${\displaystyle \mathrm {rad} (M)=\sum \{S\mid S{\mbox{ is a superfluous submodule of M}}\}\,}$

These definitions have direct dual analogues for soc(M).

## Properties

• In addition to the fact rad(M) is the sum of superfluous submodules, in a Noetherian module rad(M) itself is a superfluous submodule.
• A ring for which rad(M) ={0} for every right R module M is called a right V-ring.
• For any module M, rad(M/rad(M)) is zero.
• M is a finitely generated module if and only if M/rad(M) is finitely generated and rad(M) is a superfluous submodule of M.