Let $A$ be a commutative ring with $1_A$. Set $X=Spec(A)$. For $x\in X$, let $j_x$ be the corresponding prime ideal.
I have already understood and proved the following claims:
1) $A-j_x$ is a multiplicative set of $A$ at $j_x$.
Then, let $A_x$ be the localization of $A$ with the multiplicative set $A-j_x$. Also I proved:
2) $A_x$ is a local ring with the unique maximal ideal:
$$M_x=A_x - \{ \frac{a}{s} \vert a \in j_x \ \text{and} \ s\in A-j_x \}$$
I also verified: 3) $M_x=j_xA_x$.
Using the universal property of fraction fields, now I have tried to prove that:
$$A_x/M_x \cong Frac(A/j_x)$$
I couldn't have done yet. Any help or suggestion?