In mathematics, in the theory of moduless, 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

be a ring-module. A submodule is called maximal or cosimple if the quotient is a simple module. The radical is the intersection,

Equivalently,

.

Properties

• is the sum of superfluous submodules, in a Noetherian module itself is a superfluous submodule.

is finitely generated is finitely generated.

• is called a right V-ring.

• is zero.

• is a finitely generated module if and only if the cosocle.

See also

• Socle (mathematics)

• Jacobson radical

References