Meaning of the notation $M_p$

Question

I saw the definition of the support of a module over a commutative ring $A$ which is

the set of all prime ideals $p$ such that $M_p\ne0$.

What is $M_p$?

Answer 1

From Atiyah's commutative algebra:

Let $p$ be a prime ideal of $A$. Then $S = A \setminus p$ is multiplicatively closed. Define a relation $\sim$ on $M \times S$ as follows: $$(m,s) \sim (m',n') \iff \exists t \in S \ni t(sm'-s'm)=0$$ Then $\sim $ is an equivalence relation on $M \times S$.Let $m/s$ denote the equivalence class of the pair $(m, s )$ and let $S^{-1}M$ denote all such equivalence classes. Then $S^{-1}M$ is a module over $S^{-1}A$. We write $M_p$ instead of $S^{-1}M$ when $S=A \setminus p$