Cyclic module

In mathematics, more specifically in ring theory, a cyclic module is a module which is generated by one element over a ring. The concept is analogous to cyclic group, that is, a group which 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) = R x = {rx | r ∈ R} for some x in M. Similarly, a right R-module N is cyclic, if N = y R 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 which is generated by x then there exists a canonical isomorphism between M and R / Ann_Rx, where Ann_Rx denotes the annihilator of x in R.

See also

• Finitely generated module

References

• 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 Pub. Co., pp. 147–149, ISBN 978-0-201-55540-0, Zbl 0848.13001