Cyclic module
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.
Contents
1 Definition
2 Examples
3 Properties
4 See also
5 References
Definition
A left R-module M is called cyclic if M can be generated by a single element i.e. M = (x) = Rx = rx for some x in M. Similarly, a right R-module N is cyclic, if N = yR for some 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 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 / AnnRx, where AnnRx denotes the annihilator of x in R.
See also
- Finitely generated module
References
^ Bourbaki, Algebra I: Chapters 1–3, p. 220.mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
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
