Noncommutative rational identities are formally studied in
As explained in the Cohn’s book, one can define the division ring of noncommutative rational functions on a given alphabet.
Given a commutative field $k$ (_field of constants_), and a finite set $X = \{x_1,\ldots,x_n\}$, consider the free algebra on $X$ (in the sense of universal algebra) of signature $\{ +_2, \cdot_2, (\,)^{-1}_1, -_1\}$ and with constants in $k$ and denote it by $\mathcal{R}(X,k)$. Expressions like $0^{-1}$ and $(x-x)^{-1}$ are in $\mathcal{R}(X,k)$ as no relations are imposed (one starts with terms which are elements of $X\cup k$ and continues by nesting sequence of algebraic operations eventually connecting all terms). Let $R$ be an associative $k$-algebra, and $\phi : X\to R$ a map of sets. Then there is a subset $\mathcal{R}_\phi \subset \mathcal{R}(X,k)$ and a map $\phi_* : \mathcal{R}_\phi\to R$ uniquely determined by the rules
(constants evaluate) If $c \in k$ then $c \in \mathcal{R}_\phi$ and $\phi_*(c) = c$.
(variables evaluate) If $x \in X$ then $x \in \mathcal{R}_\phi$ and $\phi_*(x) = \phi(x)$.
(sums, products and negatives of evaluables evaluate) If $f, g \in\mathcal{R}_\phi$, then $f+g, f\cdot g, -f \in\mathcal{R}_\phi$, $\phi_*(f+g) = \phi_*(f) + \phi_*(g)$, $\phi_*(f\cdot g) = \phi_*(f)\cdot\phi_*(g)$ and $\phi_*(-f) = -\phi_*(f)$.
If $g \in \mathcal{R}_\phi$, and $\phi_*(g)$ is invertible in $R$, then $g^{-1}\in \mathcal{R}_\phi$ and $\phi_*(g^{-1}) = \phi_*(g)^{-1}$.
For every $f \in \mathcal{R}(X,k)$ define $\mathrm{Dom}_\phi\,f$ to be the set of all $|X|$-tuples $\vec{r} = (r_1,\ldots,r_{|X|}) \in R^{|X|}$ such that $f \in \mathcal{R}_\phi$ where $\phi = \phi_{\vec{r}}$ satisfies $\phi(x_i) = r_i$ for $i = 1,\ldots, |X|$. Those $f$ for which $\mathrm{Dom}_\phi f \neq \emptyset$ are called nondegenerate. It is clear that $f \notin\mathcal{R}_\phi$ iff there is a subexpression in $f$ of the form $(f_0)^{-1}$ where $f_0 \in \mathcal{R}_\phi$ and $\phi_*(f_0)$ is not invertible in $R$.
of matrices over noncommutative rings_, Funct. Anal. Appl. 25 (1991), no.2, pp. 91–102; engl. transl. 21 (1991), pp. 51–58; A theory of noncommutative determinants and characteristic functions of graphs, Funct. Anal. Appl. 26 (1992), no.4, pp. 231–246.
It is used in the study of skewfields, Cohn localization, quasideterminants, noncommutative integrable systems and so on.
Natalia Iyudu, Stanislav Shkarin, A proof of the Kontsevich periodicity conjecture, arxiv/1305.1965
M. Kontsevich, Noncommutative identities, writeup of the 2011 MPIM Bonn Arbeitstagung talk, arxiv/1109.2469