Cyclic module

In mathematics, more specifically in ring theory, a '''cyclic module''' or '''monogenous module'''<ref>{{citation|author=Bourbaki|title=Algebra I: Chapters 1–3|page=220|url=https://books.google.com/books?id=STS9aZ6F204C&pg=PA220}}</ref> 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. {{nowrap|1=''M'' = (''x'') = ''Rx'' = {''rx'' {{!}} ''r'' ∈ ''R''}{{null}}}} for some ''x'' in ''M''. Similarly, a right ''R''-module ''N'' is cyclic if {{nowrap|1=''N'' = ''yR''}} for some {{nowrap|''y'' ∈ ''N''}}.

== Examples ==

* 2'''Z''' 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.<ref>{{harvnb|Anderson|Fuller|1992|loc=Just after Proposition 2.7.}}</ref> * 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 {{nowrap|''F''[''x''] / (''x'' − ''λ'')^''n''}}; 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 {{nowrap|''R'' / Ann_''R'' ''x''}}, where {{nowrap|Ann_''R'' ''x''}} denotes the annihilator of ''x'' in ''R''.

== See also == * Finitely generated module

== References == {{Reflist}} * {{citation |last1=Anderson |first1=Frank W. |last2=Fuller |first2=Kent R. |title=Rings and categories of modules |series=Graduate Texts in Mathematics |volume=13 |edition=2 |publisher=Springer-Verlag |place=New York |year=1992 |pages=x+376 |isbn=0-387-97845-3 |mr=1245487 |doi=10.1007/978-1-4612-4418-9}} * {{cite book | author=B. Hartley | authorlink=Brian Hartley |author2=T.O. Hawkes | title=Rings, modules and linear algebra | url=https://archive.org/details/ringsmodulesline00hart | url-access=limited | publisher=Chapman and Hall | year=1970 | isbn=0-412-09810-5 | pages=[https://archive.org/details/ringsmodulesline00hart/page/n86 77], 152}} * {{Lang Algebra|edition=3|pages=147–149}}

Category:Module theory