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.

Let ${\displaystyle R}$ be a ring and ${\displaystyle M}$ a left ${\displaystyle R}$-module. A submodule ${\displaystyle N}$ of ${\displaystyle M}$ is called maximal or cosimple if the quotient ${\displaystyle M/N}$ is a simple module. The radical of the module ${\displaystyle M}$ is the intersection of all maximal submodules of ${\displaystyle M}$,

${\displaystyle \mathrm {rad} (M)=\bigcap \,\{N\mid N{\mbox{ is a maximal submodule of }}M\}}$

Equivalently,

${\displaystyle \mathrm {rad} (M)=\sum \,\{S\mid S{\mbox{ is a superfluous submodule of }}M\}}$

These definitions have direct dual analogues for ${\displaystyle \mathrm {soc} (M)}$.

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

In fact, if ${\displaystyle M}$ is finitely generated over a ring, then ${\displaystyle \mathrm {rad} (M)}$ itself is a superfluous submodule. This is because any proper submodule of ${\displaystyle M}$ is contained in a maximal submodule of ${\displaystyle M}$ when ${\displaystyle M}$ is finitely generated.

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