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 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,
Equivalently,
These definitions have direct dual analogues for soc.

Properties

In addition to the fact that rad is the sum of superfluous submodules, in a Noetherian module, rad itself is a superfluous submodule.

In fact, if M is finitely generated over a ring, then rad 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.

A ring for which rad = for every right R-module M is called a right V-ring.
For any module M, rad is zero.M is a finitely generated module if and only if the cosocle M/rad is finitely generated and rad is a superfluous submodule of M.