Cyclic module

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