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

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

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

Properties
In fact, if M is finitely generated over a ring, then rad(M) itself is a superfluous submodule. This is because any proper submodule of M is contained in a maximal submodule of M when M is finitely generated.
 * 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 the cosocle M/rad(M) is finitely generated and rad(M) is a superfluous submodule of M.