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

See also

References