$R$ is an integral domain. For $x,y\in R$, with $x,y \neq0$, prove that $\dfrac{(x)}{(xy)} \cong \dfrac{R}{(y)}$ as $R$-modules.

Do we need to find a mapping and the kernel and then use the first isomorphism theorem?



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Yes, you can use the fundamental theorem of isomorphism.

In order to do this consider the following composition of maps: $R\to(x)$ given by $a\mapsto ax$, and $(x)\twoheadrightarrow (x)/(xy)$, the canonical projection. The first map is an isomorphism, and the second is surjective, so their composition is surjective. The kernel is $\{a\in R:ax\in(xy)\}$ and you can finish the proof.


Consider the exact sequence $$ 0 \to (y) \to R \to R/(y) \to 0. $$ and note that the map $x:R \to xR$ given by multiplication by $x$ is an isomorphism of $R$-modules.

Apply this map to each term in the sequence. What do you get?


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