I'm trying to show that the homomorphic image of a principal ideal ring is again a principal ideal ring. Let $\displaystyle \phi:R \to S$. If we take a general principal ideal $\displaystyle \langle a \rangle$ and an element of this ideal, $\displaystyle ra$, then we have $\displaystyle \phi(ra) = \phi(r)\phi(a)$, but this doesn't prove it unless $\displaystyle \phi$ is surjective, correct? Because if it's not surjective then we don't know that every element in S is of the form $\displaystyle \phi(r)$, right? This is what I'm stuck on. Any hints (not answers)?