Radical of a module

Remarks

Example: let $M = \Z/12\Z$ as a $\Z$-module. Its maximal submodules are $2\Z/12\Z$ and $3\Z/12\Z$, so $\operatorname{rad}(M) = 6\Z/12\Z$, which has $2$ elements.

Non-example of a proper radical: the $\Z$-module $\Q$ has no maximal submodule, because every quotient $\Q/N$ with $N \ne \Q$ is divisible and so is not simple. So $\operatorname{rad}(\Q) = \Q$.

For $M = R$ as a left module over itself, $\operatorname{rad}(R)$ is the Jacobson radical $J(R)$. Equivalently, $\operatorname{rad}(M)$ is the intersection of the kernels of all module homomorphisms from $M$ to simple modules. For every $M$ one has $J(R) M \subseteq \operatorname{rad}(M)$, with equality when $R$ is left Artinian.