Radical of a module

Let $R$ be a ring and $M$ a left $R$-module. A submodule $N$ of M is called cosimple if $M/N$ is a simple module. The radical of the module $M$ is then the intersection of all cosimple submodules of $M.$ Mathematically, we write this as:

${\mbox{rad}}M=\bigcap _{M/N{\mbox{ is simple}}}N.$

See also

• Socle (mathematics)