Neha Patil (Editor)

Radical of a module

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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.

Contents

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,

r a d ( M ) = ⋂ { N ∣ N  is a maximal submodule of M }

Equivalently,

r a d ( M ) = ∑ { S ∣ S  is a superfluous submodule of M }

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

Properties

  • In addition to the fact rad(M) is the sum of superfluous submodules, in a Noetherian module rad(M) itself is a superfluous submodule.
  • A ring for which rad(M) ={0} for every right R module M is called a right V-ring.
  • For any module M, rad(M/rad(M)) is zero.
  • M is a finitely generated module if and only if M/rad(M) is finitely generated and rad(M) is a superfluous submodule of M.

    • References

    Radical of a module Wikipedia
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