What the difference between $A/m$ and $A_0$

Question

$A$ is a integral domain.

$m$ is a maximal ideal .

$A_0$ is the localization of $A$ by $A-0$.(Field of fractions)

$A/m$ is the quotient at $m$.

What the difference between $A/m$ and $A_0$?

Answer 1

Here's an example that will hopefully illustrate the difference. Let $A=\mathbb Z$ and $m=(5)$. $A_0=\mathbb Q$, and $A/m=\mathbb Z/5\mathbb Z$, the integers modulo 5.

Answer 2

Consider the case of $A = \mathbb{R}[x]$ and $m=(x^2+1)$. This ideal is maximal since $A/m \cong \mathbb{C}$, while $A_0 = R(x)$, the field of rational functions over $\mathbb{R}$. These are quite distinct. The first is an algebraic extension of the subfield $\mathbb{R}$, while the second is transcendental over $\mathbb{R}$.