If $f:A\rightarrow R$ be a ring homomorphism, where $A$ and $R$ are commutative rings. If $f$ is surjective and $P$ is a prime ideal in $A$, how to prove that $f(P)$ is a prime ideal in $R$?
This a an exercise I came across while self-studying joseph rotman's book advanced modern algebra. I searched this website and find similar questions but they have another condition: $P$ contains the kernel of $f$. However, I want to know how to prove this result without that condition. Give thanks to any useful help！