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 be a ring and a left -module. A submodule of is called maximal or cosimple if the quotient is a simple module. The radical of the module is the intersection of all maximal submodules of ,
Equivalently,
These definitions have direct dual analogues for .
Properties In addition to the fact that is the sum of superfluous submodules, in a Noetherian module, itself is a superfluous submodule. In fact, if is finitely generated over a ring, then itself is a superfluous submodule. This is because any proper submodule of is contained in a maximal submodule of when is finitely generated. A ring for which for every right -module is called a right V-ring. For any module , is zero. is a finitely generated module if and only if the cosocle is finitely generated and is a superfluous submodule of .
See also Socle (mathematics) Jacobson radical
References