Isomorphism between quotient of ring and principal ideals Let

Question

Isomorphism between quotient of ring and principal ideals
Let R be a principal ideal domain and suppose that

u, u^{'} \in R

are such that

\frac{R}{(u)} \tilde{=} \frac{R}{(u^{'})}

as R-modules, where (u) denotes the ideal generated by u. Is it true that

u = α u^{'}

, where

α \in R

is invertible?

Answer 1

Step 1
If

\frac{R}{(u)} ≃ ︎ \frac{R}{(u^{'})}

as R-modules, then they have the same annihilator (why?), so

(u) = (u^{'})

and thus

u = α u^{'}

with

α \in R

invertible.
In this answer R is an integral domain.

Answers (1)