Editing Ring (mathematics) (section)

=== 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}}