I've been studying localizations, and was given the next excercise:
Let $k$ be a field with characteristic 0. Let $R=k[x]$ consider the module $M=R/(x^4-x^2)$ and $U=\{1,x,x^2,\dots\}$. Compute $M[U^{-1}]$.
I'm quite lost. I know the definition for $M[U^{-1}]$, in other words they are the classe given by pair $(m,u)$ with $m\in M,u\in U$ and two pairs $(m,u),(n,v)$ are considere equivalent if exist $s\in U$ such that $s(un-mv)=0$ in $M$. A class is usually denoted $\frac{m}{u}$
I believe that I may be missing somethig respect $M$ to be able to compute $M[U^{-1}]$. If my understanding is correct $M$ $ax^3+bx+c$ with operations $mod (x^4-x^2)$.
Is there a formal way to compute $M[U^{-1}]$ or how should I approach this kind of excercise.
