Cyclic module

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title: "Cyclic module" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["module-theory"] topic_path: "general/module-theory" source: "https://en.wikipedia.org/wiki/Cyclic_module" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In mathematics, more specifically in ring theory, a cyclic module or monogenous module 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. for some x in M. Similarly, a right R-module N is cyclic if 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.
  • 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.

References

References

  1. Bourbaki. "Algebra I: Chapters 1–3".
  2. {{harvnb. Anderson. Fuller. 1992

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module-theory