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",
Equivalently,
These definitions have direct dual analogues for soc("M").