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 a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-module) that is generated by one element.
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.
Every simpleR-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.
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.
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, ISBN0-387-97845-3, MR 1245487
B. Hartley; T.O. Hawkes (1970). Rings, modules and linear algebra. Chapman and Hall. pp. 77, 152. ISBN0-412-09810-5.
cyclic, module, mathematics, more, specifically, ring, theory, cyclic, module, monogenous, module, module, over, ring, that, generated, element, concept, generalization, notion, cyclic, group, that, abelian, group, module, that, generated, element, contents, d. 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 a generalization of the notion of a cyclic group that is an Abelian group i e Z module that is generated by one element Contents 1 Definition 2 Examples 3 Properties 4 See also 5 ReferencesDefinition editA 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 edit2Z 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 l n there may also be other cyclic submodules with different annihilators see below Properties editGiven 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 editFinitely generated moduleReferences edit Bourbaki Algebra I Chapters 1 3 p 220 Anderson amp Fuller 1992 Just after Proposition 2 7 Anderson amp 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 Retrieved from https en wikipedia org w index php title Cyclic module amp oldid 1116474562, wikipedia, wiki, book, books, library,
