What does $R_I$ for a ring $R$ and ideal $I$ In this paper, I am confused by the third sentence of the third paragraph, which reads

If $A \subset B$ is a minimal ring extension, it follows from [2 (sic)] that there exists a unique maximal ideal $M$ of $A$ (called the crucial maximal ideal of $A\subset B$) such that the canonical injective ring homomorphism $A_M \to B_M (:= B_{A\setminus M})$ can be viewed as a minimal ring extension while the canonical ring homomorphism $A_P \to B_P$ is an isomorphism for all prime ideals $P$ of $A$ except $M$.

I do not know what the notation $A_M$ denotes. Can someone provide a definition?
For context, two sentences before, they state

Recall (cf. [2]) that a ring extension $A\subset B$ is a minimal ring extension if there does not exist a ring properly contained between $A$ and $B$.

I am assuming that $A\subset B$ is a ring extension if $A$ is isomorphic to a proper subring of $B$.
For completeness, here is their citation for [2]:

D. Ferrand and J.-P. Olivier, “Homomorphismes minimaux d’anneaux,” Journal of Algebra, vol. 16, pp. 461–471, 1970.

 A: A subset $S\subseteq R$ (of a commutative unital ring $R$) is called multiplicative if $a,b\in S\implies ab\in S$, $1\in S$ and $0\notin S$. For such a subset we can define the localization (mentioned by Edward Evans in the comments) at $S$:
$$S^{-1}R=\left\{\left.\frac rs\ \right|\ r\in R,s\in S\right\}$$
This construction is related to the field of fractions of an integral domain where we choose $S=R\setminus\{0\}$. The fraction $\frac rs=:(r,s)$ I wrote is formally an equivalence class w.r.t. to the equivalence relation
$$(r_1,s_1)\sim(r_2,s_2)\ \iff \ \exists s\in S:\ s(r_1s_2-r_2s_1)=0\ .$$
The localization naturally admits a ring structure (defined similar to the case of $\mathbb Q$ as field of fractions of $\mathbb Z$) and a universal property. The details may be found elsewhere.
If given a prime ideal $\mathfrak p<R$ we write $S_{\mathfrak p}=R\setminus\mathfrak p$. This subset is multiplicative as $\mathfrak p$ is prime:
$$ a\in S_{\mathfrak p},b\in S_{\mathfrak p}\iff a\notin\mathfrak p,b\notin\mathfrak p\implies ab\notin\mathfrak p\iff ab\in S_{\mathfrak p}$$
So we can define the localization $S_{\mathfrak p}^{-1}R$ which is typically denoted by $R_{\mathfrak p}$. As maximal ideals are prime the same construction can be carried out for maximal ideals $\mathfrak m<R$ and we write $R_{\mathfrak m}$. For an appropriate ideal $I<R$ we also refer tp $R_I$ to as localization at $I$.
