In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, more specifically in ring theory

Ring theory

In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...

, a cyclic module is a module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 over a ring which is generated by one element. The term is by analogy with cyclic group

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

s, that is groups which are 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
  Cyclic group
  In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
  
   is a cyclic Z-module.
• Every simple
  Simple module
  In mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M...
  
   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 ideal
  Principal ideal
  In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
  
  s as a ring. The same holds for R as a right R-module, mutatis mutandis
  Mutatis mutandis
  Mutatis mutandis is a Latin phrase meaning "by changing those things which need to be changed" or more simply "the necessary changes having been made"....
  
  .
• 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
  Annihilator (ring theory)
  In mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.-Definitions:Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M...
  
  of x in R.