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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?

Finally, where should I go to read about topics like this?

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 $\mathfrak{a}$.

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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-Alvarez Feb 16 '14 at 22:54
If $D$ satisfies the Ore condition, then you can form a ring of fractions. 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: – J. Gaddis Feb 20 '14 at 21:40

1 Answer 1

up vote 3 down vote accepted

As already noted in the comments, embedding of noncommutative domains in general into division rings is fraught with peril:

  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)

  2. The domain may not embed into any division ring at all.

The necessary and sufficient condition for a domain to be "left dense" in a division ring is the left Ore condition, and of course there is an analogous result on the right side.

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.

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