I will report on the results of recent joint work with Tobias Mai and Sheng Yin.

For a tuple of operators in a finite von Neumann algebra we consider the division closure of those operators in the affiliated unbounded operators. We address the question when this division closure is a skew field (division ring) and when it is the free skew field. We show that the first property is equivalent to the strong Atiyah property and that the second property can be characterized in terms of the non-commutative distribution of the operators. Those results also have consequences for the question of atoms in the distribution of rational functions in free variables or in the asymptotic eigenvalue distribution of matrices over polynomials in asymptotically free random matrices.