Kaplansky - commutative rings p.60 theorem 90
Let $R$ be an integral domain. Then, $R$ is Noetherian and of Krull dimension $\leq 1$ iff for any nonzero ideal $I$ of $R$, $R/I$ has a composition series.
The author says that this is an immediate corollary of the fact that [Noetherian + Krull-dimension $0$ iff Artinian] but how?
If $R$ is Noetherian and has dimension $\leq 1$, then $R/I$ is Noetherian and $0$-dimensional for any nonzero ideal $I\subset R$, so $R/I$ is Artinian and has a composition series. Conversely, suppose $R/I$ has a composition series for any nonzero ideal $I$. Then such an $R/I$ is Artinian, and so is Noetherian and $0$-dimensional. It follows that $R$ has dimension $\leq 1$ (if $P$ were a nonzero nonmaximal prime of $R$, $R/P$ would have dimension $>0$). To prove $R$ is Noetherian, let $J\subset R$ be any nonzero ideal and let $I$ be the ideal generated by a single nonzero element of $J$. Then $R/I$ is Noetherian, so $J/I$ is finitely generated. Since $I$ is finitely generated, this implies $J$ is finitely generated.
