cyclic module

Definition

Let $M$ be an R-module. $M$ is called cyclic if it is generated by a single element $m\in M$.

Relation to ideals of ring

$M$ is cyclic if and only if it is isomorphic to $R/I$ for some left ideal $I\subset R$.

Proof

Assume $M$ is cyclic, so $M=⟨m⟩$.
Then we can look at the $R$-module homomorphism

$$\begin{array}{rl}f:R& ⟶M\\ r& ⟼rm\end{array}$$

since $M=⟨m⟩$, this is surjective. Thus, by the first isomorphism theorem we have

$$M\cong R/\ker f.$$

Note that $\ker f={\text{Ann}}_R(M)$, and we know that ${\text{Ann}}_R(M)$ is a left ideal, we have that $M\cong R/I$.

Next assume that $M\cong R/I$ for some ideal $I\subset R$. We know that $R/I$ is generated by the element $[1]$ since for each $[r]\in R/I$,

$$r\cdot [1]=[r]$$

Therefore, since we have an isomorphism $\phi :R/I\to M$, then we have $M=⟨\phi ([1])⟩$.