Universal skew field of fractions

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.

Tags:

Category Theory

Universal Property

Noncommutative Algebra

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