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
  1. Bourbaki, Algebra I: Chapters 1–3, p. 220
  2. Anderson & Fuller, Just after Proposition 2.7.
  3. Anderson & Fuller, Proposition 2.7.
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