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 $R’$.
Add to solve later Tagged: preimage
Add to solve later The Preimage of Prime ideals are Prime Ideals
Problem 412
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 later Group Homomorphism, Preimage, and Product of Groups
Problem 208
Let $G, G’$ be groups and let $f:G \to G’$ be a group homomorphism.
Put $N=\ker(f)$. Then show that we have
\[f^{-1}(f(H))=HN.\]
Add to solve later The Preimage of a Normal Subgroup Under a Group Homomorphism is Normal
Problem 116
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 $G$.
Add to solve later 
