# Generators and presentations of a division ring

Let $$D$$ be a division ring/algebra. I'm confused about what are the generators of $$D$$.

On one hand, the multiplicative group $$D^\times$$ is generated by some generators $$S$$ as a group. Most algebra textbooks carefully defines this process. But at this point, we have all the elements of $$D$$ other than zero.

On the other hand, there is a general categorical result asserting every ring is the quotient of a free ring, which is the $$\mathbb{Z}$$-polynomials of words over some set, with juxtaposition as the multiplication. Say the generating set is $$S$$ and the free ring is $$FR(S)$$. Then $$D\simeq FR(S)/I$$, where $$I$$ is some ideal.

The difference of these two generating processes occurs at the inverses. In the group generating process, $$S^{-1}$$ is simply a set with the same cardinality as $$S$$. Whereas, in the quotient of free ring construction, I'm not sure how the inverses are defined from $$S$$.

Can someone please guide me through the quotient ring method, and show how the inverses are created? Book reference is welcome too. Thanks in advance.

• Thanks for the answer. After some thought, is it correct to say for $D\simeq FR(S)/I$, it is equivalent to finding a maximal ideal $I$ that is also a maximal left/right ideal? May 15 at 13:22
• And, in general, if $R\simeq FR(S)/I$, where $\pi: FR(S)\to R$ is the canonical projection, then $u\in R^\times$ is a unit iff for every $u^*\in \pi^{-1}(u)$, the subring of the free ring $\langle\{1\}\cup \{u^*\}\cup I\rangle$ has no proper one-sided ideals containing $I$ May 15 at 17:17