I'm reading a book about Commutative Algebra from my library and I'm stuck in a problem about localization. Here I put the statement and my attemps.
Let $M$ be a maximal ideal of a domain $R$ and let $S=R\setminus M$. Prove that there is an epimorphism $\varphi: S^{-1}R\rightarrow R/M$ such that $\varphi(a/1)=a+M$ whose kernel is $S^{-1}M$.
My first attemp was to define $\varphi$ exactly like the last conclusion, but the problem comes from to see the fact about the kernel, concretely, the inclusion $S^{-1}M\subset \ker\varphi$, because this leads to a "contradiction" of the choice of $\varphi$ since I've no idea which is the image of the elements of the form $r/s$ in general.
My nexts attemps were by try to find such epimorphism using "brute force", for example, the function whose sends $a/b$ to $a+b, a+b-1$ or to $a$ but at least I have one propertie not satisfied in the definition of homomorphism.
Honestly, I tried this exercise arround three hours and I don't see how to solve it. I will appreciate any hint.
Regards, thanks
If $m/s\in S^{-1}M$, then you can express it in $S^{-1}R$ as the product $$ \frac{m}{s}=\frac{m}{1}\cdot\left(\frac{s}{1}\right)^{-1}. $$
Let $\bar{a}=a+M$ in the field $R/M$. Applying $\varphi$ to the above equation, $$\varphi(m/s)=\bar{m}\cdot\bar{s}^{-1}=\bar{0}\cdot\bar{s}^{-1}=\bar{0},$$
so $S^{-1}M\subseteq\ker\varphi$.