# An explicit construction of the universal division ring of fractions of $E\langle\langle x_1,\ldots, x_d\rangle \rangle$

### Andrei Jaikin-Zapirain

Universidad Autónoma de Madrid, Spain

## Abstract

We give a sufficient and necessary condition for a regular Sylvester matrix rank function on a ring $R$ to be equal to its inner rank $\rho_R$. We apply it in two different contexts.

In our first application, we reprove a recent result of T. Mai, R. Speicher and S. Yin: if $X_1,\ldots, X_d$ are operators in a finite von Neumann algebra $\mathcal M$ with a faithful normal trace $\tau$, then they generate the free division ring on $X_1,\ldots, X_d$ in the algebra of unbounded operators affiliated to $\mathcal M$ if and only if the space of tuples $(T_1,\ldots, T_d)$ of finite rank operators on $L^2(\mathcal M,\tau)$ satisfying

is trivial.

In our second and main application we construct explicitly the universal division ring of fractions of $E\langle\langle x_1,\ldots, x_n\rangle\rangle$, where $E$ is a division ring, and we use it in order to show the following instance of pro-$p$ Lück approximation.

Let $F$ be a finitely generated free pro $p$-group, $F=F_1 > F_2 > \cdots$ a chain of normal open subgroups of $F$ with trivial intersection and $A$ a matrix over $\mathbb F_p [[F]]$. Denote by $A_i$ the matrix over $\mathbb F_p[F/F_i]$ obtained from the matrix $A$ by applying the natural homomorphism $\mathbb F_p [[F]] \to \mathbb F_p[F/F_i]$. Then there exists the limit

and it is equal to the inner rank $\rho_{\mathbb F_p [[F]]}(A)$ of the matrix $A$.

## Cite this article

Andrei Jaikin-Zapirain, An explicit construction of the universal division ring of fractions of $E\langle\langle x_1,\ldots, x_d\rangle \rangle$. J. Comb. Algebra 4 (2020), no. 4, pp. 369–395

DOI 10.4171/JCA/47