# 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*,

Equivalently,

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 the cosocle*M*/rad(*M*) is finitely generated and rad(*M*) is a superfluous submodule of*M*.

## See also

## References

- Alperin, J.L.; Rowen B. Bell (1995).
*Groups and representations*. Springer-Verlag. p. 136. ISBN 0-387-94526-1. - Anderson, Frank Wylie; Kent R. Fuller (1992).
*Rings and Categories of Modules*. Springer-Verlag. ISBN 978-0-387-97845-1.

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