| 释义 |
- Definition
- Examples
- Properties
- See also
- References
In mathematics, more specifically in ring theory, a cyclic module or monogenous module[1] is a module that is generated by one element over a ring. The concept is analogous to cyclic group, that is, a group 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 - Every cyclic group is a cyclic Z-module.
- Every simple R-module M is a cyclic module since the submodule generated by any nonzero element x of M is necessarily the whole module M.
- 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 - Finitely generated module
References 1. ^{{citation|author=Bourbaki|title=Algebra I: Chapters 1–3|page=220|url=https://books.google.com/books?id=STS9aZ6F204C&pg=PA220}}
- {{cite book | author=B. Hartley | authorlink=Brian Hartley |author2=T.O. Hawkes | title=Rings, modules and linear algebra | publisher=Chapman and Hall | year=1970 | isbn=0-412-09810-5 | pages=77,152}}
- {{Lang Algebra|edition=3|pages=147–149}}
1 : Module theory |