Suppose we have a homomorphism $\phi: A \rightarrow B$ where $A$ and $B$ are rings. $\phi$ is injective and not necessarily surjective. Is image of prime ideal and maximal ideals in $A$ also prime and maximal in $B$?
My attempt: Suppose $P$ is prime ideal of $A$ then $\phi(a)\phi(b) \in \phi(P)$ then $ab \in P$ and hence $a \in P$ or $b \in P$ which implies $\phi(a) \in P$ or $\phi(b) \in P$.
Is this proof correct? Also how can proceed for maximal ideal.