Editing Ring (mathematics) (section)

=== Central simple algebra and Brauer group ===
{{main|Central simple algebra}}

For a field {{mvar|k}}, a {{mvar|k}}-algebra is central if its center is {{mvar|k}} and is simple if it is a [[simple ring]]. Since the center of a simple {{mvar|k}}-algebra is a field, any simple {{mvar|k}}-algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a {{mvar|k}}-algebra. The matrix ring of size {{mvar|n}} over a ring {{mvar|R}} will be denoted by {{math|''R''{{sub|''n''}}}}.

The [[Skolem–Noether theorem]] states any automorphism of a central simple algebra is inner.

Two central simple algebras {{mvar|A}} and {{mvar|B}} are said to be ''similar'' if there are integers {{mvar|n}} and {{mvar|m}} such that <math>A \otimes_k k_n \approx B \otimes_k k_m.</math>{{sfnp|Milne|CFT|loc=Ch IV, §2|ps=}} Since <math>k_n \otimes_k k_m \simeq k_{nm},</math> the similarity is an equivalence relation. The similarity classes {{math|[''A'']}} with the multiplication <math>[A][B] = \left[A \otimes_k B\right]</math> form an abelian group called the [[Brauer group]] of {{mvar|k}} and is denoted by {{math|Br(''k'')}}. By the [[Artin–Wedderburn theorem]], a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring.

For example, {{math|Br(''k'')}} is trivial if {{mvar|k}} is a finite field or an algebraically closed field (more generally [[quasi-algebraically closed field]]; cf. [[Tsen's theorem]]). <math>\operatorname{Br}(\R)</math> has order 2 (a special case of the [[Frobenius theorem (real division algebras)|theorem of Frobenius]]). Finally, if {{mvar|k}} is a nonarchimedean [[local field]] (for example, {{nowrap|{{tmath|\Q _p}}),}} then <math>\operatorname{Br}(k) = \Q /\Z </math> through the [[Hasse invariant of an algebra|invariant map]].

Now, if {{mvar|F}} is a field extension of {{mvar|k}}, then the base extension <math>- \otimes_k F</math> induces {{math|Br(''k'') → Br(''F'')}}. Its kernel is denoted by {{math|Br(''F'' / ''k'')}}. It consists of {{math|[''A'']}} such that <math>A \otimes_k F</math> is a matrix ring over {{mvar|F}} (that is, {{mvar|A}} is split by {{mvar|F}}.) If the extension is finite and Galois, then {{math|Br(''F'' / ''k'')}} is canonically isomorphic to <math>H^2\left(\operatorname{Gal}(F/k), k^*\right).</math>{{sfnp|Serre|1950|ps=}}

[[Azumaya algebra]]s generalize the notion of central simple algebras to a commutative local ring.