Universal skew field of fractions

Let $R$ be a non-commutative ring with unity. A skew field $K$ containing $R$ is called a universal skew field of fractions of $R$ if

• it is generated by $R$ as a skew field (i.e. there is no proper subskewfield of $K$ containing $R$)
• for any skew field $L$ and any morphism of rings $\varphi : R \to L$ there exists a subring $K_0$ of $K$ with $R \subseteq K_0$ and a morphism of rings $\theta : K_0 \to L$ extending $\varphi$ and with the property that $$ \forall x \in K_0 (x^{-1} \in K_0 \Leftrightarrow \theta(x) \neq 0). $$

This is the definition I saw in a noncommutative geometry course. We went on to examine a few examples, but a lot of things are still not clear to me:

• Why would the 'naive' generalisation of the universal property in the commutative case not give the right notion in the non-commutative case?
• In what category is the above definition of a universal skew field of fractions actually an initial object? Don't we need uniqueness of $K_0$ and $\varphi$?
• Because why else would the universal skew field of fractions be unique up to isomorphism?

Note that uniqueness does not follow from the fact that $R$ generates $K$. An example from my course (sketch):

Let $k$ be a commutative field, $k<x_1, x_2>$ the polynomial ring in two non-commuting variables. For every $n \in \mathbb{N}$, $n \geq 2$ consider the skew polynomial ring $k[t][X, n]$, in which $tX = Xt^n$. This is a right Ore ring, hence can be embedded into a skew field. Also one has an embedding $$ \phi_n : k < x_1, x_2 > \to k[t][X, n] $$ by setting $\phi_n(x_1) = X$ and $\phi_n(x_2) = Xt$, but the fields generated by $\phi_n(k < x_1, x_2 >)$ and $\phi_m(k < x_1, x_2 >)$ are not isomorphic if $n \neq m$, as one has $$ \phi_n(x_1)^{-1}\phi_n(x_2)\phi_n(x_1) = X^{-1}XtX = tX = Xt^n = Xtt^{n-1} = Xt(X^{-1}Xt)^{n-1} = \phi_n(x_2)(\phi_n(x_1)\phi_n(x_2))^{n-1} $$ and this depends on the choice of $n$!

Solution 1:

The example you give shows why the 'naive' notion doesn't work for non-commutative rings in general. The free algebra $R=k\langle x_1,x_2\rangle$ has several different skew fields of fractions. But any map between skew fields of fractions of $R$ must be an isomorphism, so there could only be a "universal" (in the naive sense) skew field of fractions if there were a unique skew field of fractions.

The right category in which to take the initial object is pretty much described by the second part of your definition of the universal skew field of fractions:

If $R$ is a ring then the skew fields of fractions of $R$ form a category where a morphism from $R\hookrightarrow K$ to $R\hookrightarrow L$ is a "specialization"; i.e., a (necessarily surjective) ring homomorphism $\beta: K_0\to L$ from a subring $K_0$ of $K$ with $R\subseteq K_0$ with the property that $\beta$ is the identity on $R$ and $\ker(\beta)$ is precisely the set of non-units of $K_0$. The universal skew field of fractions is the initial object in this category.

Solution 2:

Complement The category of Jeremy is ``unfolded'' w.r.t. that of P. M. Cohn (in Free Ideal Rings and Localization in General Rings) and it is also my way (because associativity is, IMHO, easier to write down completely). You recover Cohn's category (in the book above) by passing to quotient (the relation between two `unfolded specializations'' $\beta_i, i=1,2$ being that there is a common subring $K_{0i}, i=1,2$ where they coincide). It is an easy exercise to prove that, $F$ being the quotient functor, the initial points of Cohn are precisely the images of Jeremy's (hence essentially the same) by $F$.