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In [[mathematics]], in the theory of [[Module (mathematics)|module]]s, the '''radical''' of a module is a component in the theory of structure and classification. It is a generalization of the [[Jacobson radical]] for [[ring theory|ring]]s. In many ways, it is the [[duality (mathematics)|dual]] notion to that of the [[socle (mathematics)|socle]] soc(''M'') of ''M''. | |||
==Definition== | |||
Let ''R'' be a [[Ring (mathematics)|ring]] and ''M'' a left ''R''-module. A submodule ''N'' of ''M'' is called '''[[maximal submodule|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'', | |||
:<math>\mathrm {rad}(M) = \bigcap \{ N \mid N \mbox{ is a maximal submodule of M} \} \,</math> | |||
Equivalently, | |||
:<math>\mathrm {rad}(M) = \sum \{ S \mid S \mbox{ is a superfluous submodule of M} \} \,</math> | |||
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''. | |||
==See also== | |||
*[[Socle (mathematics)]] | |||
*[[Jacobson radical]] | |||
==References== | |||
* {{cite book | last = Alperin | first = J.L. | author-link=J. L. Alperin | coauthors = Rowen B. Bell | title = Groups and representations | publisher = [[Springer-Verlag]] | year = 1995 | isbn = 0-387-94526-1 | pages = 136 }} | |||
* {{cite book | last=Anderson | first=Frank Wylie | coauthors=Kent R. Fuller | title=Rings and Categories of Modules | publisher=[[Springer-Verlag]] | isbn=978-0-387-97845-1 | year=1992}} | |||
[[Category:Module theory]] | |||
{{Abstract-algebra-stub}} | |||
Revision as of 20:58, 15 January 2014
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,
Equivalently,
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.
See also
References
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