Division rings

#ring_theory

Definition and examples

Definition of division ring

A division ring is a ring in which every nonzero element is a unit.

A community division ring is a field.

Examples still to come.

An interesting result

Here's a little result:

Abelian subgroups of division rings

Suppose $R$ is a division ring and $A \leq R^{\times}$ is an abelian subgroup of the multiplicative group $R^{\times}$ . Then there exists a field $F$ and an isomorphism of $A$ with a subgroup of $F^{\times}$ .

Proof. First consider the subset $D \subseteq R$ defined by

D = {\sum_{a \in A} n_{a} a ∣ n_{a} \in Z, all but finitely many zero} .

As sets, we certainly have $A \subseteq D$ . Also, the subset $D$ is easily seen to be a commutative subring of the division ring $R$ . While it might not be true that $D$ is itself a division ring, at the very least $D$ is an integral domain. Let $F$ be a field of fractions of $D$ , with injective ring morphism $i : D \to F$ . Restricting $i$ to $A$ yields an injective group morphism $i : A \to F^{\times}$ , providing an isomorphism of $A$ with a subgroup of $F^{\times}$ . $◼$

Division subring generated by

A

The above result is mentioned in a parenthetical hint in Exercise 14 of Section 18.1 in Dummit & Foote. In that hint, it is suggested that the reader "consider the division subring generated by $A$ ." I'm not quite certain of the validity of this suggestion, though. At the very least, I see no reason to believe that the subring $D$ , above, is a division subring. (Perhaps there is some result I'm missing?) If we "expand" $D$ to the smallest division subring containing $D$ (assuming such a subring exists), it's no longer clear such a subring is commutative.

Corollay

If $R$ is a division ring and $A \leq R^{\times}$ is a finite abelian subgroup of $R^{\times}$ , then $A$ is cyclic.

Proof. By the previous result, there exists a field $F$ such that $A$ is isomorphic to a (finite, abelian) subgroup $B \leq F^{\times}$ . Since every finite subgroup of the group of units of a field is cyclic, it follows that $B$ (and hence $A$ ) is cyclic. $◼$

Categorical interpretation

From the point of view of category theory, the above results (and their proofs) are a complete mess. We start with a division ring, then talk about a subgroup of the group of units of that division ring, then back to a subring generated by the subgroup (considered as a subset), then to a field with a ring morphism, which we then restrict to a group morphism. Blech! We should really try to carefully frame the situation with relevant categories and the functors we're using to move between them.

Coming soon.