I have $R=K[x]$, $M=K[x]/(x^2)$, $U=K[x]\setminus(x)$ and I want to find $U^{-1}M$. I have issues understanding exactly what set is $U$. Is $U \cong K$? Because that was my first guess. Now, I think $U^{-1}M\cong M$ but i'm not sure if my reasons are right.
2026-03-27 15:35:35.1774625735
The set $K[x]\setminus (x)$ and localization
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The ideal $(x)$ contains all polynomials that are a multiple of $x$. Thus $U$ contains all polynomials that have a non-zero constant term. Does this help you?