cyclic module
In mathematics, more specifically in ring theory, a cyclic module or monogenous module{{citation|author=Bourbaki|title=Algebra I: Chapters 1–3|page=220|url=https://books.google.com/books?id=STS9aZ6F204C&pg=PA220}} 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
- 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.{{harvnb|Anderson|Fuller|1992|loc=Just after Proposition 2.7.}}
- 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 / AnnR x}}, where {{nowrap|AnnR x}} denotes the annihilator of x in R.
See also
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}}