infinity of abelian submonoid

Question

Are all abelian submonoids of $\mathbb Z$ infinite ?

I would say yes, because a submonoid of $\mathbb Z$ is a monoid itself and so it can be infinite.

Is that correct?

Answer 1

The answer is "almost all". Of course, $\{0\} \subseteq\mathbf Z$ is a finite submonoid. But: If $M\subseteq \mathbf Z$ is any submonoid containing an element $m \ne 0$. Then $\{\sum_{i=1}^k m : k \in \mathbf N \}$ is an infinite subset of $M$. Hence $M$ is infinite.