Editing Ring theory (section)

==Structures and invariants of rings==

===Dimension of a commutative ring===
{{main|Dimension theory (algebra)}}
In this section, ''R'' denotes a commutative ring.  The [[Krull dimension]] of ''R'' is the supremum of the lengths ''n'' of all the chains of prime ideals <math>\mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_n</math>. It turns out that the polynomial ring <math>k[t_1, \cdots, t_n]</math> over a field ''k'' has dimension ''n''. The fundamental theorem of dimension theory states that the following numbers coincide for a noetherian local ring <math>(R, \mathfrak{m})</math>:<ref>{{harvnb|Matsumura|1989|loc=Theorem 13.4}}</ref>
*The Krull dimension of ''R''.
*The minimum number of the generators of the <math>\mathfrak{m}</math>-primary ideals.
*The dimension of the graded ring <math>\textstyle \operatorname{gr}_{\mathfrak{m}}(R) = \bigoplus_{k \ge 0} \mathfrak{m}^k/{\mathfrak{m}^{k+1}}</math> (equivalently, 1 plus the degree of its [[Hilbert polynomial]]).

A commutative ring ''R'' is said to be [[Catenary ring|catenary]] if for every pair of prime ideals <math>\mathfrak{p} \subset \mathfrak{p}'</math>, there exists a finite chain of prime ideals <math>\mathfrak{p} = \mathfrak{p}_0 \subsetneq \cdots \subsetneq \mathfrak{p}_n = \mathfrak{p}'</math> that is maximal in the sense that it is impossible to insert an additional prime ideal between two ideals in the chain, and all such maximal chains between <math>\mathfrak{p}</math> and  <math>\mathfrak{p}'</math> have the same length.  Practically all noetherian rings that appear in applications are catenary.  Ratliff proved that a noetherian local integral domain ''R'' is catenary if and only if for every prime ideal <math>\mathfrak{p}</math>,
:<math>\operatorname{dim}R = \operatorname{ht}\mathfrak{p} + \operatorname{dim}R/\mathfrak{p}</math>
where <math>\operatorname{ht}\mathfrak{p}</math> is the [[Height (ring theory)|height]] of <math>\mathfrak{p}</math>.<ref>{{harvnb|Matsumura|1989|loc=Theorem 31.4}}</ref>

If ''R'' is an integral domain that is a finitely generated ''k''-algebra, then its dimension is the [[transcendence degree]] of its field of fractions over ''k''. If ''S'' is an [[integral extension]] of a commutative ring ''R'', then ''S'' and ''R'' have the same dimension.

Closely related concepts are those of [[depth (ring theory)|depth]] and [[global dimension]]. In general, if ''R'' is a noetherian local ring, then the depth of ''R'' is less than or equal to the dimension of ''R''. When the equality holds, ''R'' is called a [[Cohen–Macaulay ring]]. A [[regular local ring]] is an example of a Cohen–Macaulay ring. It is a theorem of Serre that ''R'' is a regular local ring if and only if it has finite global dimension and in that case the global dimension is the Krull dimension of ''R''. The significance of this is that a global dimension is a [[homological algebra|homological]] notion.

===Morita equivalence===
{{main|Morita equivalence}}
Two rings ''R'', ''S'' are said to be [[Morita equivalent]] if the category of left modules over ''R'' is equivalent to the category of left modules over ''S''. In fact, two commutative rings which are Morita equivalent must be isomorphic, so the notion does not add anything new to the [[category theory|category]] of commutative rings. However, commutative rings can be Morita equivalent to noncommutative rings, so Morita equivalence is coarser than isomorphism. Morita equivalence is especially important in algebraic topology and functional analysis.

===Finitely generated projective module over a ring and Picard group===
Let ''R'' be a commutative ring and <math>\mathbf{P}(R)</math> the set of isomorphism classes of finitely generated [[projective module]]s over ''R''; let also <math>\mathbf{P}_n(R)</math> subsets consisting of those with constant rank ''n''. (The rank of a module ''M'' is the continuous function <math>\operatorname{Spec}R \to \mathbb{Z}, \, \mathfrak{p} \mapsto \dim M \otimes_R k(\mathfrak{p})</math>.<ref>{{harvnb|Weibel|2013|loc=Ch I, Definition 2.2.3}}</ref>) <math>\mathbf{P}_1(R)</math> is usually denoted by Pic(''R''). It is an abelian group called the [[Picard group]] of ''R''.<ref>{{harvnb|Weibel|2013|loc=Definition preceding Proposition 3.2 in Ch I}}</ref> If ''R'' is an integral domain with the field of fractions ''F'' of ''R'', then there is an exact sequence of groups:<ref>{{harvnb|Weibel|2013|loc=Ch I, Proposition 3.5}}</ref>
:<math>1 \to R^* \to F^* \overset{f \mapsto fR}\to \operatorname{Cart}(R) \to \operatorname{Pic}(R) \to 1</math>
where <math>\operatorname{Cart}(R)</math> is the set of [[fractional ideal]]s of ''R''. If ''R'' is a [[Regular ring|regular]] domain (i.e., regular at any prime ideal), then Pic(R) is precisely the [[divisor class group]] of ''R''.<ref>{{harvnb|Weibel|2013|loc=Ch I, Corollary 3.8.1}}</ref>

For example, if ''R'' is a principal ideal domain, then Pic(''R'') vanishes. In algebraic number theory, ''R'' will be taken to be the [[ring of integers]], which is Dedekind and thus regular. It follows that Pic(''R'') is a finite group ([[finiteness of class number]]) that measures the deviation of the ring of integers from being a PID.<!-- discuss coordinate ring -->

One can also consider the [[group completion]] of <math>\mathbf{P}(R)</math>; this results in a commutative ring K<sub>0</sub>(R). Note that K<sub>0</sub>(R) = K<sub>0</sub>(S) if two commutative rings ''R'', ''S'' are Morita equivalent.

{{See also|Algebraic K-theory}}

===Structure of noncommutative rings===
{{main|Noncommutative ring}}

The structure of a [[noncommutative ring]] is more complicated than that of a commutative ring. For example, there exist [[Simple ring|simple]] rings that contain no non-trivial proper (two-sided) ideals, yet contain non-trivial proper left or right ideals. Various invariants exist for commutative rings, whereas invariants of noncommutative rings are difficult to find. As an example, the [[nilradical of a ring]], the set of all nilpotent elements, is not necessarily an ideal unless the ring is commutative. Specifically, the set of all nilpotent elements in the ring of all {{nowrap|''n'' × ''n''}} matrices over a division ring never forms an ideal, irrespective of the division ring chosen. There are, however, analogues of the nilradical defined for noncommutative rings, that coincide with the nilradical when commutativity is assumed.

The concept of the [[Jacobson radical]] of a ring; that is, the intersection of all right (left) [[Annihilator (ring theory)|annihilators]] of [[Simple module|simple]] right (left) modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right (left) ideals in the ring, shows how the internal structure of the ring is reflected by its modules. It is also a fact that the intersection of all maximal right ideals in a ring is the same as the intersection of all maximal left ideals in the ring, in the context of all rings; irrespective of whether the ring is commutative.

Noncommutative rings are an active area of research due to their ubiquity in mathematics. For instance, the ring of ''n''-by-''n'' [[Matrix (mathematics)|matrices over a field]] is noncommutative despite its natural occurrence in [[geometry]], [[physics]] and many parts of mathematics. More generally, [[endomorphism ring]]s of abelian groups are rarely commutative, the simplest example being the endomorphism ring of the [[Klein four-group]].

One of the best-known strictly noncommutative ring is the [[quaternions]].