cyclic module

In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-module) that is generated by one element.

Definition

A left ''R''-module ''M'' is called cyclic if ''M'' can be generated by a single element i.e. for some ''x'' in ''M''. Similarly, a right ''R''-module ''N'' is cyclic if for some .

Examples

* 2Z as a Z-module is a cyclic module.
* In fact, every cyclic group is a cyclic Z-module.
* Every simple ''R''-module ''M'' is a cyclic module since the submodule generated by any non-zero element ''x'' of ''M'' is necessarily the whole module ''M''. In general, a module is simple if and only if it is nonzero and is generated by each of its nonzero elements.
* If the ring ''R'' is considered as a left module over itself, then its cyclic submodules are exactly its left principal ideals as a ring. The same holds for ''R'' as a right ''R''-module, mutatis mutandis.
* If ''R'' is ''F'' 'x'' the ring of polynomials over a field ''F'', and ''V'' is an ''R''-module which is also a finite-dimensional vector space over ''F'', then the Jordan blocks of ''x'' acting on ''V'' are cyclic submodules. (The Jordan blocks are all isomorphic to ; there may also be other cyclic submodules with different annihilators; see below.)

Properties

* Given a cyclic ''R''-module ''M'' that is generated by ''x'', there exists a canonical isomorphism between ''M'' and , where denotes the annihilator of ''x'' in ''R''.

See also

* Finitely generated module

References

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* {{Lang Algebra, edition=3, pages=147–149

Module theory