You want to show that the homomorphic image is a PID. Every function is a surjection onto its image.
I'm trying to show that the homomorphic image of a principal ideal ring is again a principal ideal ring. Let . If we take a general principal ideal and an element of this ideal, , then we have , but this doesn't prove it unless is surjective, correct? Because if it's not surjective then we don't know that every element in S is of the form , right? This is what I'm stuck on. Any hints (not answers)?