Definition:Division Ring

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

Definition 1

$\forall x \in R_{\ne 0_R}: \exists! x^{-1} \in R_{\ne 0_R}: x^{-1} \circ x = x \circ x^{-1} = 1_R$

where $R^*$ denotes the set of elements of $R$ without the ring zero $0_R$:

$R_{\ne 0_R} = R \setminus \set {0_R}$

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 division ring as the definition of a field.

However, this is non-standard: it is usually specified that a field product has to be commutative.

Some sources refer to a division ring as a skew field, but the latter is usually applied to a division ring whose ring product is specifically non-commutative.

Also see

  • Results about division rings can be found here.