Editing Ring (mathematics) (section)

=== Division ring ===
A [[division ring]] is a ring such that every non-zero element is a unit. A commutative division ring is a [[field (mathematics)|field]]. A prominent example of a division ring that is not a field is the ring of [[quaternion]]s. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every ''finite'' domain (in particular finite division ring) is a field; in particular commutative (the [[Wedderburn's little theorem]]).

Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field.

The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the [[Cartan–Brauer–Hua theorem]].<!--
need more context. To give a concrete example, we consider a division ring {{mvar|D}} whose center {{mvar|k}} is the field <math>\Q _p</math> of {{mvar|p}}-adic rational numbers (or more generally a nonarchimedean local field).{{sfnp|Milne|CFT|loc=Ch IV, §4|ps=}} The valuation of {{mvar|k}} uniquely extends to any subfield of {{mvar|D}}. Since {{mvar|D}} is a union of subfields, we thus obtain the valuation {{mvar|v}} of {{mvar|D}}. (This has to be verified; it is not a priori obvious that {{mvar|v}} is indeed a valuation). Define
<math display="block">O_D = \{ x \in D \mid v(x) \ge 0 \}, \quad \mathfrak{P} = \{ x \in D \mid v(x) > 0 \}.</math>
Then <math>O_D</math> is a subring of {{mvar|D}} with the unique maximal ideal that is <math>\mathfrak{P}.</math> It is called the ring of integers of {{mvar|D}}.
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A [[cyclic algebra]], introduced by [[L. E. Dickson]], is a generalization of a [[quaternion algebra]].