Thread: Homomorphic image of principal ideal ring

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)?

You want to show that the homomorphic image is a PID. Every function is a surjection onto its image.

φ:R-->φ(R) is always onto.

Yes, true. So do we then have the desired result since subrings are preserved, and so that the elements of the image are of the form with ?