 Homomorphic images of Noetherian rings are Noetherian

Let $\varphi :R\to S$ be a surjective ring homomorphism and let $R$ be a Noetherian ring. For every ideal $J$ of $S$, its inverse image is a finitely generated ideal of $R$. If ${\varphi }^{-1}(J)=(a_1,\dots ,a_r)$, then $J=(\varphi (a_1),\dots ,\varphi (a_r))$. Thus $S$ is a Noetherian ring. In particular, every quotient ring of a Noetherian ring is a Noetherian ring. For a ring homomorphism that is not surjective, apply the same argument with its image as codomain.