# Smallest skew-field containing a non-commutative ring.

Let $R$ be an integral domain and take $D = R - \left\{ 0 \right\}$. The ring $D^{-1}R$ is the smallest field containing $R$ as a subring.

Now suppose that I have a non-commutative ring $N$. Suppose also that $N$ contains no zero divisors. How do I go about finding the smallest skew-field that contains $N$? What happens if we remove the restriction that $N$ has no zero divisors? I would think that there is no longer a skew-field containing $N$, but is there a "closest approximation" to one?

I should add that my motivation is that I am trying to bolster my intuition about ideals $\mathfrak{a}$ of general rings. If I could find the smallest skew-field $S$ containing $N$, then perhaps I can see more about the structure of the ideals of $\mathfrak{a}$ by looking at the left (or right) action of $S - \left\{0\right\}$ on the elements of $\mathfrak{a}$.

• Sometimew, even when there are no zero divisors, there is no such skew field. This is a classical subject: you should pick any of the standard textbooks on non commutative rings, like McConnel and Robson, which devotes its fisrst chapter to rings of fractions. – Mariano Suárez-Álvarez Feb 16 '14 at 22:54
• If $D$ satisfies the Ore condition, then you can form a ring of fractions. en.wikipedia.org/wiki/Ore_condition Without the Ore condition, $N$ may be contained in multiple non-isomorphic minimal skew fields, or $N$ may not be contained in any skew fields. – Jack Schmidt Feb 17 '14 at 0:32
• A good free resource on this topic is a set of notes by Allen Bell: pantherfile.uwm.edu/adbell/www/Research/locnotes.pdf – J. Gaddis Feb 20 '14 at 21:40

1. The domain may not "densely" embed into a division ring (this is what the condition "everything in the field is of the form $rs^{-1}$ for some $r,s\in D$" amounts to in commutative fields of fractions)
For reading on this, I'd like to recommend T.Y. Lam's Lectures on Modules and Rings chapter 4 ("Rings of quotients") section $9$ (subsections "the good", "the bad" and "the ugly" give examples of what can go wrong) and section $10$ ("Classical rings of quotients") covers what you are interested in.