- Source: 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,
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{\displaystyle \mathrm {rad} (M)=\bigcap \,\{N\mid N{\mbox{ is a maximal submodule of }}M\}}
Equivalently,
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{\displaystyle \mathrm {rad} (M)=\sum \,\{S\mid S{\mbox{ 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.
In fact, if M is finitely generated over a ring, then rad(M) itself is a superfluous submodule. This is because any proper submodule of M is contained in a maximal submodule of M when M is finitely generated.
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 the cosocle M/rad(M) is finitely generated and rad(M) is a superfluous submodule of M.
See also
Socle (mathematics)
Jacobson radical
References
Alperin, J.L.; Rowen B. Bell (1995). Groups and representations. Springer-Verlag. p. 136. ISBN 0-387-94526-1.
Anderson, Frank Wylie; Kent R. Fuller (1992). Rings and Categories of Modules. Springer-Verlag. ISBN 978-0-387-97845-1.