Radical of a module

In many ways, it is the dual notion to that of the socle soc(M) of M. 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(M).

In fact, if M is finitely generated over a ring, then rad(M) itself is a superfluous submodule.

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