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.

Definition[edit]

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,

\mathrm {rad}(M) = \bigcap \{ N \mid N \mbox{ is a maximal submodule of M} \} \,

Equivalently,

\mathrm {rad}(M) = \sum \{ S \mid S \mbox{ is a superfluous submodule of M} \} \,

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

Properties[edit]

  • 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.

See also[edit]

References[edit]