# Finitely generated subrings of $\mathbb{R}$ are finitely approximable

In Ivanov's Finite Approximability of Modular Teichmüller Groups, for the proof of Lemma 2, the following is stated:

Let $G$ be a finitely generated group and $\tau: G \to \operatorname{PSL}(2,\mathbb{R})$ be a homomorphism. [...] Since $G$ is finitely generated, $\tau(G) \subset \operatorname{PSL}(2,A)$ for some finitely generated ring $A \subset \mathbb{R}$. It is well known that such a ring is finitely approximable, i.e. for any $d \in A$, $d \neq 0$, there is an ideal $J$ such that $A/J$ is finite and $d \notin J$.

The reference given for this is:

Yu. I. Merzlyakov, Rational Groups [in Russian], Nauka, Moscow (1980)

Unfortunately, I am unable to find the original article or a translation, Google only comes up with other articles and textbooks using the same reference.

Could anyone point me to the article in question or maybe a different article where I could look up a proof for this fact?

EDIT: It has been pointed out to me that the original Russian version can be found by searching for "Merzlyakov gruppy Nauka 1980 djvu". Since I don't understand Russian, I was hoping to find a translation, or a different reference for the fact stated.

• It's now called "residually finite". "Finitely approximable" is used in other meanings.
– YCor
Apr 25, 2019 at 6:15

$A$ is a finitely generated integral domain. By the Nullstellensatz, its Jacobson radical vanishes (because its nilradical vanishes), meaning every nonzero element $a \in A$ avoids some maximal ideal, say $m$. By the Nullstellensatz again, $A/m$ is a finite field.
This argument shows more generally that any finitely generated linear group (subgroup of $GL_n(K)$ for $K$ a field) is residually finite, which is Malcev's theorem.