Exercise 111.1.1. Let $A$ be a ring, and ${\mathfrak m}$ a maximal ideal. In $A[X]$ let $\tilde{\mathfrak m}_1 = ({\mathfrak m}, X)$ and $\tilde{\mathfrak m}_2 = ({\mathfrak m}, X-1)$. Show that

\[ A[X]_{\tilde{\mathfrak m}_1} \cong A[X]_{\tilde{\mathfrak m}_2}. \]

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