Editing Ring (mathematics) (section)

== Special kinds of rings ==

=== Domains ===
A [[zero ring|nonzero]] ring with no nonzero [[zero-divisor]]s is called a [[domain (ring theory)|domain]]. A commutative domain is called an [[integral domain]]. The most important integral domains are principal ideal domains, PIDs for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a [[unique factorization domain]] (UFD), an integral domain in which every nonunit element is a product of [[prime element]]s (an element is prime if it generates a [[prime ideal]].) The fundamental question in [[algebraic number theory]] is on the extent to which the [[ring of integers|ring of (generalized) integers]] in a [[number field]], where an "ideal" admits prime factorization, fails to be a PID.

Among theorems concerning a PID, the most important one is the [[structure theorem for finitely generated modules over a principal ideal domain]]. The theorem may be illustrated by the following application to linear algebra.{{sfnp|Lang|2002|loc=Ch XIV, §2|ps=}} Let {{mvar|V}} be a finite-dimensional vector space over a field {{mvar|k}} and {{math|''f'' : ''V'' → ''V''}} a linear map with minimal polynomial {{mvar|q}}. Then, since {{math|''k''[''t'']}} is a unique factorization domain, {{mvar|q}} factors into powers of distinct irreducible polynomials (that is, prime elements):
<math display="block">q = p_1^{e_1} \ldots p_s^{e_s}.</math>

Letting <math>t \cdot v = f(v),</math> we make {{mvar|V}} a {{math|''k''[''t'']}}-module. The structure theorem then says {{mvar|V}} is a direct sum of [[cyclic module]]s, each of which is isomorphic to the module of the form <math>k[t] / \left(p_i^{k_j}\right).</math> Now, if <math>p_i(t) = t - \lambda_i,</math> then such a cyclic module (for {{mvar|p{{sub|i}}}}) has a basis in which the restriction of {{mvar|f}} is represented by a [[Jordan matrix]]. Thus, if, say, {{mvar|k}} is algebraically closed, then all {{mvar|p{{sub|i}}}}'s are of the form {{math|''t'' – ''λ{{sub|i}}''}} and the above decomposition corresponds to the [[Jordan canonical form]] of {{mvar|f}}.
[[File:Ringhierarchy.png|thumb|272x272px|Hierarchy of several classes of rings with examples.]]
In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a [[regular local ring]]. A regular local ring is a UFD.{{sfnp|Weibel|2013|p=[https://books.google.com/books?id=Ja8xAAAAQBAJ&pg=PA26 26]|loc=Ch 1, Theorem 3.8|ps=}}

The following is a chain of [[subclass (set theory)|class inclusions]] that describes the relationship between rings, domains and fields:
{{Commutative ring classes}}

=== 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}}.
-->

A [[cyclic algebra]], introduced by [[L. E. Dickson]], is a generalization of a [[quaternion algebra]].

=== Semisimple rings ===
{{Main|Semisimple module}}

A ''[[semisimple module]]'' is a direct sum of simple modules.  A ''[[semisimple ring]]'' is a ring that is semisimple as a left module (or right module) over itself.

====Examples====
* A [[division ring]] is semisimple (and [[simple ring|simple]]).
* For any division ring {{mvar|D}} and positive integer {{mvar|n}}, the matrix ring {{math|M{{sub|''n''}}(''D'')}} is semisimple (and [[simple ring|simple]]).
* For a field {{mvar|k}} and finite group {{mvar|G}}, the group ring {{math|''kG''}} is semisimple if and only if the [[characteristic (algebra)|characteristic]] of {{mvar|k}} does not divide the [[order (algebra)|order]] of {{mvar|G}} ([[Maschke's theorem]]).
* [[Clifford algebra]]s are semisimple.

The [[Weyl algebra]] over a field is a [[simple ring]], but it is not semisimple.  The same holds for a [[differential operator#Ring of multivariate polynomial differential operators|ring of differential operators in many variables]].

====Properties====
Any module over a semisimple ring is semisimple. (Proof: A free module over a semisimple ring is semisimple and any module is a quotient of a free module.)

For a ring {{mvar|R}}, the following are equivalent:
* {{mvar|R}} is semisimple.
* {{mvar|R}} is [[artinian ring|artinian]] and [[semiprimitive ring|semiprimitive]].
* {{mvar|R}} is a finite [[direct product]] <math display="inline"> \prod_{i=1}^r \operatorname{M}_{n_i}(D_i) </math> where each {{math|''n''{{sub|''i''}}}} is a positive integer, and each {{math|''D''{{sub|''i''}}}} is a division ring ([[Artin–Wedderburn theorem]]).

Semisimplicity is closely related to separability. A unital associative algebra {{mvar|A}} over a field {{mvar|k}} is said to be [[separable algebra|separable]] if the base extension <math>A \otimes_k F</math> is semisimple for every [[field extension]] {{math|''F'' / ''k''}}. If {{mvar|A}} happens to be a field, then this is equivalent to the usual definition in field theory (cf. [[separable extension]].)

=== 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.

=== Valuation ring ===
{{main|Valuation ring}}

If {{mvar|K}} is a field, a [[valuation (algebra)|valuation]] {{mvar|v}} is a group homomorphism from the multiplicative group {{math|''K''{{sup|∗}}}} to a totally ordered abelian group {{mvar|G}} such that, for any {{math|''f''}}, {{math|''g''}} in {{mvar|K}} with {{math|''f'' + ''g''}} nonzero, {{math|''v''(''f'' + ''g'') ≥ min{''v''(''f''), ''v''(''g'')}.}} The [[valuation ring]] of {{mvar|v}} is the subring of {{mvar|K}} consisting of zero and all nonzero {{mvar|f}} such that {{math|''v''(''f'') ≥ 0}}.

Examples:
* The field of [[formal Laurent series]] <math>k(\!(t)\!)</math> over a field {{mvar|k}} comes with the valuation {{mvar|v}} such that {{math|''v''(''f'')}} is the least degree of a nonzero term in {{mvar|f}}; the valuation ring of {{mvar|v}} is the [[formal power series ring]] <math>k[\![t]\!].</math>
* More generally, given a field {{mvar|k}} and a totally ordered abelian group {{mvar|G}}, let <math>k(\!(G)\!)</math> be the set of all functions from {{mvar|G}} to {{mvar|k}} whose supports (the sets of points at which the functions are nonzero) are [[well ordered]]. It is a field with the multiplication given by [[convolution]]: <math display="block">(f*g)(t) = \sum_{s \in G} f(s)g(t - s).</math> It also comes with the valuation {{mvar|v}} such that {{math|''v''(''f'')}} is the least element in the support of {{mvar|f}}. The subring consisting of elements with finite support is called the [[group ring]] of {{mvar|G}} (which makes sense even if {{mvar|G}} is not commutative). If {{mvar|G}} is the ring of integers, then we recover the previous example (by identifying {{mvar|f}} with the series whose {{mvar|n}}th coefficient is&nbsp;{{math|''f''(''n'')}}.)

{{See also|Novikov ring|uniserial ring}}