Definition:Division Ring
Definition
A division ring is a ring with unity $\struct {R, +, \circ}$ with the following properties:
Definition 1
- $\forall x \in R^*: \exists! x^{-1} \in R^*: x^{-1} \circ x = x \circ x^{-1} = 1_R$
where $R^*$ denotes the set of elements of $R$ without the ring zero $0_R$:
- $R^* = R \setminus \set {0_R}$
That is, every non-zero element of $R$ has a (unique) non-zero product inverse.
Definition 2
Definition 3
- $R$ has no proper elements.
Also known as
Some sources use this as the definition of a field, although this is non-standard: it is usually specified that a field product has to be commutative.
Also see
- Results about division rings can be found here.
Sources
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- 1944: Emil Artin and Arthur N. Milgram: Galois Theory (2nd ed.) (translated by Arthur N. Milgram) ... (next): $\text I$. Linear Algebra: $\text A$. Fields
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields