Cohn localization examples I'm working on my master's thesis, part of which involves an exposition on Cohn localization. (nlab discussion)
In Free ideal rings and localization in general rings, Sec 7.4, Cohn gives a construction for a ring $\Sigma^{-1}R$. Given a set of matrices $\Sigma$ (with a mild closure condition), this ring admits a homomorphism from $R$ which is universal with respect to the property that the image of each matrix in $\Sigma$ is invertible over $\Sigma^{-1}R$.
I understand the construction and its uses in finding conditions for embeddability of domains into skew fields and the existence of universal fields, but I would really like to have some concrete examples of what the construction actually gives.
There are a few trivial examples - if $R$ is commutative, then $\Sigma^{-1}R$ is just the ring of quotients of $R$ with denominator set comprised of the determinants of matrices in $\Sigma$. If $\Sigma$ contains the zero matrix, the Cohn localization is the zero ring.
But neither of these highlights what makes Cohn localization a novel idea or sheds any light on what "matrix inverting homomorphisms" look like away from the commutative case.
Cohn's book also lacks examples. Where else can I see some concrete and informative examples of the ring $\Sigma^{-1}R$?
 A: Cohn localization of group ring of free group on $r$ generators $k[F_r]$ w.r.t. set of matrices which are invertible after augmentation $\epsilon$ is the ring of "noncommutative rational functions", if $k$ is PID. To define what that "rational functions" are, recall that free group ring is embedded into formal series ring $\Gamma_r := k\langle\langle x_1, \dots, x_r \rangle \rangle$ via Magnus homomorphism 
$$\mu: k[F_r] \to k\langle\langle x_1, \dots, x_r \rangle \rangle; \, x_i \mapsto 1 + x_i + x_i^2 + \dots.$$
Now consider set of $(\epsilon, Id)$-derivations $\delta_i$ on $\Gamma_r$ which send $x_iw$ to $w$ and $x_jw$ to $0$. They and their composites constitute a ring of operators $D := k[\delta_1, \dots, \delta_r]$ acting on $\Gamma_r$ Now, a rational function is an element $s$ of $\Gamma_r$ for which $Ds$ is a finitely generated $k$-module. One can readily check that the set of rational functions is closed under multiplication; if $r = 1$ that will result in usual rational functions ring.
What makes Cohn localization "novel", as you say? Free group rings do not satisfy Ore condition, so there's no way to construct this ring via usual method of "ring of fractions".
