Is PID cyclic? I cannot understand quotient ring is cyclic

Question

In the textbook, it says “If $R$ is a principal ideal domain and $r$ is an elt of $R$, then the quotient ring $R/(r)$ is a cyclic R-module.

I dont know why $R/(r)$ is cyclic.

Thank you in advance.

Answer 1

I am just elaborating on Berci's comment.

Maybe you are confusing two different notions of being cyclic:

• A cyclic group is a group $G$, such there exists $g \in G$ with $G = \langle g \rangle$, i.e., all elements of $G$ are of the form $g^m$ for $m \in \mathbb Z$.

• A cyclic $R$-module (for a commutative unital ring $R$) is an $R$-module $M$ such that there exists an $m \in M$ with $M = \langle m \rangle_R$, i.e., $M$ is generated by $m$ as an $R$-module. By definition, $\langle m \rangle_R = \{rm \ | \ r \in R\}$.

Here, obviously, we are speaking about the notion of a cyclic $R$-module. Now, as noted in the comments, $R/(r)$ is generated by $1 + (r)$ as an $R$-module. Here, one does not need to use that $R$ is a principal ideal domain (and it's hard to guess why you require this, because you don't provide much context).