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

- 2
**Z**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*/ Ann_{R}*x*, where Ann_{R}*x*denotes the annihilator of*x*in*R*.

- Every module is a sum of cyclic submodules.
^{[3]}

## See also

## References

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

- Anderson, Frank W.; Fuller, Kent R. (1992),
*Rings and categories of modules*, Graduate Texts in Mathematics,**13**(2 ed.), New York: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3, MR 1245487 - 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, pp. 147–149, ISBN 978-0-201-55540-0, Zbl 0848.13001