Radical of a module

Summary

The radical of a module is the intersection of all its maximal submodules. It is the dual notion of the socle, which is a sum of minimal submodules. For the ring viewed as a module over itself, the radical is the Jacobson radical. It is used to pass from a module to the quotient $M/\operatorname{rad}(M)$ in many arguments, such as Nakayama's lemma.