Cyclic module

In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring which is generated by one element. The term is by analogy with cyclic groups, that is groups which are generated by one element.

Definition

A R-module M is called cyclic if M can be generated by a single element i.e. M=(x)= R x for some x in M

Examples

• Every cyclic group is a cyclic Z-module
• Every simple R-module M is a cyclic module since the submodule generated by any element x of M is necessarily the whole module M
• If R is a commutative ring, then the cyclic submodules of the R-module R are exactly its principal ideals as a ring.
• If R is F[x], the ring over 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

• cyclic group
• finitely generated module