$\mathbf{Z}$ is Noetherian

The ring $\mathbf{Z}$ of integers is Noetherian because every ideal of $\mathbf{Z}$ is generated by one element.

Proposition 4.1.9   Every ideal of the ring $\mathbf{Z}$ of integers is principal.

Proof. Suppose $I$ is a nonzero ideal in  $\mathbf{Z}$. Let $d$ the least positive element of $I$. Suppose that $a\in I$ is any nonzero element of $I$. Using the division algorithm, write $a=dq + r$, where $q$ is an integer and $0\leq r < d$. We have $r=a-dq\in I$ and $r<d$, so our assumption that $d$ is minimal implies that $r=0$, so $a=dq$ is in the ideal generated by $d$. Thus $I$ is the principal ideal generated by $d$. $\qedsymbol$

Proposition 4.1.6 and 4.1.9 together imply that any finitely generated abelian group is Noetherian. This means that subgroups of finitely generated abelian groups are finitely generated, which provides the missing step in our proof of the structure theorem for finitely generated abelian groups.