Question

Let aR is a non-zero ideal of a PID R. Prove that R/aR is a ring with finitely many ideals
Please, help

Answer 1

This is what I have so far:
Let bR is an ideal of R which contains aR, hence aR⊂bR, that is b|a or a = bm for some m in R. And we know that with a = bm, a has finitely many factors. I know I have to write some more argument to link between aR =<a> with the result of Division Algorithm above to get the final conclusion that R/aR has a finitely many ideals. But I don't know how to.

Answer 2

gooooooooooooooot it. lalalala...