Cyclic module
In mathematics, more specifically in ring theory, a cyclic module or monogenous module[1] is a module over a ring that is generated by one element. 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. M = (x) = Rx = {rx | r ∈ R} for some x in M. Similarly, a right R-module N is cyclic if N = yR for some 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.[2]
- 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 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 R / AnnR x, where AnnR x denotes the annihilator of x in R.
- Every module is a sum of cyclic submodules.[3]
See also
References
- ^ Bourbaki, Algebra I: Chapters 1–3, p. 220
- ^ Anderson & Fuller 1992, Just after Proposition 2.7.
- ^ Anderson & Fuller 1992, Proposition 2.7.
- Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, vol. 13 (2 ed.), New York: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3, MR 1245487
- B. Hartley; T.O. Hawkes (1970). Rings, modules and linear algebra. Chapman and Hall. pp. 77, 152. ISBN 0-412-09810-5.
- Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, pp. 147–149, ISBN 978-0-201-55540-0, Zbl 0848.13001