Question

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

Answer 1

This is what I have so far:
Let bR is an ideal of R which contains aR, hence \(aR\subset bR\), that is b|a or a = bm for some m in R
And we know that with a = bm , a has a countable factors. In other words, 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.
Appreciate any tips

Answer 2

Btw, since R is a PID, R is a UFD , that allows us apply Division Algorithm

Answer 3

oh, one more thing, may be it works, :)

Answer 4

Correspondence theorem gives us the injection of the maps
hence if b is a factor of a, b is finitely decomposed to irreducible factors, that is the ideals of R/aR is finitely many. ha!! I don't convince myself with that proof.