= Homomorphic images of Noetherian rings are Noetherian
Let $\phi:R\to S$ 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 $\phi^{-1}(J)=(a_1,\ldots,a_r)$, then $J=(\phi(a_1),\ldots,\phi(a_r))$. 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.
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