Definition:Division Ring

Definition

A division ring is a ring with unity $\struct {R, +, \circ}$ with one of the following equivalent 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

Every non-zero element of $R$ is a unit.

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

• Equivalence of Definitions of Division Ring

• Definition:Field (Abstract Algebra)
• Definition:Skew Field

• 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