Isomorphism between $R = \mathbb{R}[X, Y]/(X^2 + Y^2 - 1)$ and $IJ$ where $I = (x - 1, y)$ and $J = (y, x - 1)$

Question

Let $R = \mathbb{R}[X, Y]/(X^2 + Y^2 - 1)$ and $I = (x - 1, y)$, $J = (x, y - 1)$, where $x = X + (X^2 + Y^2 - 1)$ and $y = Y + (X^2 + Y^2 - 1)$. I have managed to prove that $I + J = R$ and $IJ = (x + y - 1)$. I tried to use some isomorphism theorems, but they don't seem to work. Any suggestions how to prove that $R \cong (x + y - 1) = IJ$?

Answer 1

This is true in far more generality: Let $R$ be a commutative ring, and let $x \in R$ be any element. Then we have an $R$-module homomorphism $R \to (x)$ defined for each $r \in R$ by $r \mapsto rx$. This map is surjective by the definition of $(x)$, and is injective if and only if $x$ is not a zero-divisor. In other words: any principal ideal generated by a non-zero-divisor is isomorphic as an $R$-module to $R$.