Editing Division ring (section)

== Examples ==
* As noted above, all [[Field (mathematics)|fields]] are division rings.
* The [[quaternion]]s form a noncommutative division ring.
* The subset of the quaternions {{math|''a'' + ''bi'' + ''cj'' + ''dk''}}, such that {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, and {{mvar|d}} belong to a fixed subfield of the [[real number]]s, is a noncommutative division ring. When this subfield is the field of [[rational number]]s, this is the division ring of ''rational quaternions''.
* Let <math>\sigma: \Complex \to \Complex</math> be an [[automorphism]] of the field {{nowrap|<math>\Complex</math>.}}  Let <math>\Complex((z,\sigma))</math> denote the ring of [[formal Laurent series]] with complex coefficients, wherein multiplication is defined as follows:  instead of simply allowing coefficients to commute directly with the indeterminate {{nowrap|<math>z</math>,}} for {{nowrap|<math>\alpha\in\Complex</math>,}} define <math>z^i\alpha := \sigma^i(\alpha) z^i</math> for each index {{nowrap|<math>i\in\mathbb{Z}</math>.}}  If <math>\sigma</math> is a non-trivial automorphism of [[complex number]]s (such as [[complex conjugate|the conjugation]]), then the resulting ring of Laurent series is a noncommutative division ring known as a ''skew Laurent series ring'';{{sfnp|Lam|2001|p=10}} if {{math|1=''σ'' = [[identity function|id]]}} then it features the [[ring of formal Laurent series|standard multiplication of formal series]]. This concept can be generalized to the ring of Laurent series over any fixed field {{nowrap|<math>F</math>,}} given a nontrivial {{nowrap|<math>F</math>-automorphism}} {{nowrap|<math>\sigma</math>.}}