# Isomorphism between quotient of ring and principal ideals Let

Rosa Townsend 2022-04-13 Answered
Isomorphism between quotient of ring and principal ideals
Let R be a principal ideal domain and suppose that $u,{u}^{\prime }\in R$ are such that $\frac{R}{\left(u\right)}\stackrel{\sim }{=}\frac{R}{\left({u}^{\prime }\right)}$ as R-modules, where (u) denotes the ideal generated by u. Is it true that $u=\alpha {u}^{\prime }$, where $\alpha \in R$ is invertible?
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Kaitlynn Craig
Step 1
If $\frac{R}{\left(u\right)}\simeq ︎\frac{R}{\left({u}^{\prime }\right)}$
as R-modules, then they have the same annihilator (why?), so
$\left(u\right)=\left({u}^{\prime }\right)$
and thus
$u=\alpha {u}^{\prime }$
with $\alpha \in R$
invertible.
In this answer R is an integral domain.