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Radical of a module

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Source updated March 25, 2026

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.

01Definition

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 $\mathrm {soc} (M)$ .

02Properties

In addition to the fact that $\mathrm {rad} (M)$ is the sum of superfluous submodules, in a Noetherian module, $\mathrm {rad} (M)$ itself is a superfluous submodule.

In fact, if $M$ is finitely generated over a ring, then $\mathrm {rad} (M)$ 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.

A ring for which $\mathrm {rad} (M)=\{0\}$ for every right $R$ -module $M$ is called a right V-ring.
For any module $M$ , $\mathrm {rad} (M/\mathrm {rad} (M))$ is zero.
$M$ is a finitely generated module if and only if the cosocle $M/\mathrm {rad} (M)$ is finitely generated and $\mathrm {rad} (M)$ is a superfluous submodule of $M$ .

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Sources and credits

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Radical of a module

01Definition

02Properties

Sources and credits

Related topics

Socle (mathematics)

Jacobson radical