cyclic module

Definition

Let be an R-module. is called cyclic if it is generated by a single element .

Relation to ideals of ring

is cyclic if and only if it is isomorphic to for some left ideal .

Proof

Assume is cyclic, so .
Then we can look at the -module homomorphism

since , this is surjective. Thus, by the first isomorphism theorem we have

Note that , and we know that is a left ideal, we have that .

Next assume that for some ideal . We know that is generated by the element since for each ,

Therefore, since we have an isomorphism , then we have .