## If $R$ is a Noetherian Ring and $f:R\to R’$ is a Surjective Homomorphism, then $R’$ is Noetherian

## Problem 413

Suppose that $f:R\to R’$ is a surjective ring homomorphism.

Prove that if $R$ is a Noetherian ring, then so is $R’$.

Suppose that $f:R\to R’$ is a surjective ring homomorphism.

Prove that if $R$ is a Noetherian ring, then so is $R’$.

Let $f: R\to R’$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R’$.

Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.

Add to solve laterLet $G, G’$ be groups and let $f:G \to G’$ be a group homomorphism.

Put $N=\ker(f)$. Then show that we have

\[f^{-1}(f(H))=HN.\]

Let $G$ and $G’$ be groups and let $f:G \to G’$ be a group homomorphism.

If $H’$ is a normal subgroup of the group $G’$, then show that $H=f^{-1}(H’)$ is a normal subgroup of the group $G$.