Cyclic module

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 a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-module) that is generated by one element.

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.