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 $\mathrm {soc} (M)$.

Properties

• 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 $M$.

See also

• Socle (mathematics)
• Jacobson radical

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.