In algebra, ring theory is the study of rings—algebraic
structures in which addition and multiplication
are defined and have similar properties to
those operations defined for the integers.
Ring theory studies the structure of rings,
their representations, or, in different language,
modules, special classes of rings (group rings,
division rings, universal enveloping algebras),
as well as an array of properties that proved
to be of interest both within the theory itself
and for its applications, such as homological
properties and polynomial identities.
Commutative rings are much better understood
than noncommutative ones. Algebraic geometry
and algebraic number theory, which provide
many natural examples of commutative rings,
have driven much of the development of commutative
ring theory, which is now, under the name
of commutative algebra, a major area of modern
mathematics. Because these three fields (algebraic
geometry, algebraic number theory and commutative
algebra) are so intimately connected it is
usually difficult and meaningless to decide
which field a particular result belongs to.
For example, Hilbert's Nullstellensatz is
a theorem which is fundamental for algebraic
geometry, and is stated and proved in terms
of commutative algebra. Similarly, Fermat's
last theorem is stated in terms of elementary
arithmetic, which is a part of commutative
algebra, but its proof involves deep results
of both algebraic number theory and algebraic
geometry.
Noncommutative rings are quite different in
flavour, since more unusual behavior can arise.
While the theory has developed in its own
right, a fairly recent trend has sought to
parallel the commutative development by building
the theory of certain classes of noncommutative
rings in a geometric fashion as if they were
rings of functions on (non-existent) 'noncommutative
spaces'. This trend started in the 1980s with
the development of noncommutative geometry
and with the discovery of quantum groups.
It has led to a better understanding of noncommutative
rings, especially noncommutative Noetherian
rings.For the definitions of a ring and basic
concepts and their properties, see ring (mathematics).
The definitions of terms used throughout ring
theory may be found in the glossary of ring
theory.
== Commutative rings ==
A ring is called commutative if its multiplication
is commutative. Commutative rings resemble
familiar number systems, and various definitions
for commutative rings are designed to formalize
properties of the integers. Commutative rings
are also important in algebraic geometry.
In commutative ring theory, numbers are often
replaced by ideals, and the definition of
the prime ideal tries to capture the essence
of prime numbers. Integral domains, non-trivial
commutative rings where no two non-zero elements
multiply to give zero, generalize another
property of the integers and serve as the
proper realm to study divisibility. Principal
ideal domains are integral domains in which
every ideal can be generated by a single element,
another property shared by the integers. Euclidean
domains are integral domains in which the
Euclidean algorithm can be carried out. Important
examples of commutative rings can be constructed
as rings of polynomials and their factor rings.
Summary: Euclidean domain => principal ideal
domain => unique factorization domain => integral
domain => Commutative ring.
=== Algebraic geometry ===
Algebraic geometry is in many ways the mirror
image of commutative algebra. This correspondence
started with Hilbert's Nullstellensatz that
establishes a one-to-one correspondence between
the points of an algebraic variety, and the
maximal ideals of its coordinate ring. This
correspondence has been enlarged and systematized
for translating (and proving) most geometrical
properties of algebraic varieties into algebraic
properties of associated commutative rings.
Alexander Grothendieck completed this by introducing
schemes, a generalization of algebraic varieties,
which may be built from any commutative ring.
More precisely,
the spectrum of a commutative ring is the
space of its prime ideals equipped with Zariski
topology, and augmented with a sheaf of rings.
These objects are the "affine schemes" (generalization
of affine varieties), and a general scheme
is then obtained by "gluing together" (by
purely algebraic methods) several such affine
schemes, in analogy to the way of constructing
a manifold by gluing together the charts of
an atlas.
== Noncommutative rings ==
Noncommutative rings resemble rings of matrices
in many respects. Following the model of algebraic
geometry, attempts have been made recently
at defining noncommutative geometry based
on noncommutative rings.
Noncommutative rings and associative algebras
(rings that are also vector spaces) are often
studied via their categories of modules. A
module over a ring is an abelian group that
the ring acts on as a ring of endomorphisms,
very much akin to the way fields (integral
domains in which every non-zero element is
invertible) act on vector spaces. Examples
of noncommutative rings are given by rings
of square matrices or more generally by rings
of endomorphisms of abelian groups or modules,
and by monoid rings.
=== Representation theory ===
Representation theory is a branch of mathematics
that draws heavily on non-commutative rings.
It studies abstract algebraic structures by
representing their elements as linear transformations
of vector spaces, and studies
modules over these abstract algebraic structures.
In essence, a representation makes an abstract
algebraic object more concrete by describing
its elements by matrices and the algebraic
operations in terms of matrix addition and
matrix multiplication, which is non-commutative.
The algebraic objects amenable to such a description
include groups, associative algebras and Lie
algebras. The most prominent of these (and
historically the first) is the representation
theory of groups, in which elements of a group
are represented by invertible matrices in
such a way that the group operation is matrix
multiplication.
== Some relevant theorems ==
General
Isomorphism theorems for rings
Nakayama's lemmaStructure theorems
The Artin–Wedderburn theorem determines
the structure of semisimple rings
The Jacobson density theorem determines the
structure of primitive rings
Goldie's theorem determines the structure
of semiprime Goldie rings
The Zariski–Samuel theorem determines the
structure of a commutative principal ideal
ring
The Hopkins–Levitzki theorem gives necessary
and sufficient conditions for a Noetherian
ring to be an Artinian ring
Morita theory consists of theorems determining
when two rings have "equivalent" module categories
Cartan–Brauer–Hua theorem gives insight
on the structure of division rings
Wedderburn's little theorem states that finite
domains are fieldsOther
The Skolem–Noether theorem characterizes
the automorphisms of simple rings
== 
Structures and invariants of rings ==
=== Dimension of a commutative ring ===
The Krull dimension of a commutative ring
R is the supremum of the lengths n of all
the increasing chains of prime ideals
p
0
⊊
p
1
⊊
⋯
⊊
p
n
{\displaystyle {\mathfrak {p}}_{0}\subsetneq
{\mathfrak {p}}_{1}\subsetneq \cdots \subsetneq
{\mathfrak {p}}_{n}}
. For example, the polynomial ring
k
[
t
1
,
⋯
,
t
n
]
{\displaystyle k[t_{1},\cdots ,t_{n}]}
over a field k has dimension n. The fundamental
theorem in the dimension theory states the
following numbers coincide for a noetherian
local ring
(
R
,
m
)
{\displaystyle (R,{\mathfrak {m}})}
:
The Krull dimension of R.
The minimum number of the generators of the
m
{\displaystyle {\mathfrak {m}}}
-primary ideals.
The dimension of the graded ring
gr
m
⁡
(
R
)
=
⊕
k
≥
0
m
k
/
m
k
+
1
{\displaystyle \operatorname {gr} _{\mathfrak
{m}}(R)=\oplus _{k\geq 0}{\mathfrak {m}}^{k}/{{\mathfrak
{m}}^{k+1}}}
(equivalently, one plus the degree of its
Hilbert polynomial).A commutative ring R is
said to be catenary if any pair of prime ideals
p
⊂
p
′
{\displaystyle {\mathfrak {p}}\subset {\mathfrak
{p}}'}
can be extended to a chain of prime ideals
p
=
p
0
⊊
⋯
⊊
p
n
=
p
′
{\displaystyle {\mathfrak {p}}={\mathfrak
{p}}_{0}\subsetneq \cdots \subsetneq {\mathfrak
{p}}_{n}={\mathfrak {p}}'}
of same finite length such that there is no
prime ideal that is strictly contained in
two consecutive terms. Practically all noetherian
rings that appear in application are catenary.
If
(
R
,
m
)
{\displaystyle (R,{\mathfrak {m}})}
is a catenary local integral domain, then,
by definition,
dim
⁡
R
=
ht
⁡
p
+
dim
⁡
R
/
p
{\displaystyle \operatorname {dim} R=\operatorname
{ht} {\mathfrak {p}}+\operatorname {dim} R/{\mathfrak
{p}}}
where
ht
⁡
p
=
dim
⁡
R
p
{\displaystyle \operatorname {ht} {\mathfrak
{p}}=\operatorname {dim} R_{\mathfrak {p}}}
is the height of
p
{\displaystyle {\mathfrak {p}}}
. It is a deep theorem of Ratliff that the
converse is also true.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
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
notion.
=== 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 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
P
(
R
)
{\displaystyle \mathbf {P} (R)}
the set of isomorphism classes of finitely
generated projective modules over R; let also
P
n
(
R
)
{\displaystyle \mathbf {P} _{n}(R)}
subsets consisting of those with constant
rank n. (The rank of a module M is the continuous
function
Spec
⁡
R
→
Z
,
p
↦
dim
⁡
M
⊗
R
k
(
p
)
{\displaystyle \operatorname {Spec} R\to \mathbb
{Z} ,\,{\mathfrak {p}}\mapsto \dim M\otimes
_{R}k({\mathfrak {p}})}
.)
P
1
(
R
)
{\displaystyle \mathbf {P} _{1}(R)}
is usually denoted by Pic(R). It is an abelian
group called the Picard group of R. If R is
an integral domain with the field of fractions
F of R, then there is an exact sequence of
groups:
1
→
R
∗
→
F
∗
→
f
↦
f
R
Cart
⁡
(
R
)
→
Pic
⁡
(
R
)
→
1
{\displaystyle 1\to R^{*}\to F^{*}{\overset
{f\mapsto fR}{\to }}\operatorname {Cart} (R)\to
\operatorname {Pic} (R)\to 1}
where
Cart
⁡
(
R
)
{\displaystyle \operatorname {Cart} (R)}
is the set of fractional ideals of R. If R
is a regular domain (i.e., regular at any
prime ideal), then Pic(R) is precisely the
divisor class group of R.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.
One can also consider the group completion
of
P
(
R
)
{\displaystyle \mathbf {P} (R)}
; this results in a commutative ring K0(R).
Note that K0(R) = K0(S) if two commutative
rings R, S are Morita equivalent.
=== Structure of noncommutative rings ===
The structure of a noncommutative ring is
more complicated than that of a commutative
ring. For example, there exist simple rings,
containing no non-trivial proper (two-sided)
ideals, which 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, need not be
an ideal unless the ring is commutative. Specifically,
the set of all nilpotent elements in the ring
of all n x 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
annihilators of 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;
whether commutative or noncommutative.
Noncommutative rings serve as an active area
of research due to their ubiquity in mathematics.
For instance, the ring of n-by-n matrices
over a field is noncommutative despite its
natural occurrence in geometry, physics and
many parts of mathematics. More generally,
endomorphism rings of abelian groups are rarely
commutative, the simplest example being the
endomorphism ring of the Klein four-group.
One of the best known noncommutative rings
is the division ring of quaternions.
== Applications ==
=== The ring of integers of a number field
===
=== The coordinate ring of an algebraic variety
===
If X is an affine algebraic variety, then
the set of all regular functions on X forms
a ring called the coordinate ring of X. For
a projective variety, there is an analogous
ring called the homogeneous coordinate ring.
Those rings are essentially the same things
as varieties: they correspond in essentially
a unique way. This may be seen via either
Hilbert's Nullstellensatz or scheme-theoretic
constructions (i.e., Spec and Proj).
=== Ring of invariants ===
A basic (and perhaps the most fundamental)
question in the classical invariant theory
is to find and study polynomials in the polynomial
ring
k
[
V
]
{\displaystyle k[V]}
that are invariant under the action of a finite
group (or more generally reductive) G on V.
The main example is the ring of symmetric
polynomials: symmetric polynomials are polynomials
that are invariant under permutation of variable.
The fundamental theorem of symmetric polynomials
states that this ring is
R
[
σ
1
,
…
,
σ
n
]
{\displaystyle R[\sigma _{1},\ldots ,\sigma
_{n}]}
where
σ
i
{\displaystyle \sigma _{i}}
are elementary symmetric polynomials.
== History ==
Commutative ring theory originated in algebraic
number theory, algebraic geometry, and invariant
theory. Central to the development of these
subjects were the rings of integers in algebraic
number fields and algebraic function fields,
and the rings of polynomials in two or more
variables. Noncommutative ring theory began
with attempts to extend the complex numbers
to various hypercomplex number systems. The
genesis of the theories of commutative and
noncommutative rings dates back to the early
19th century, while their maturity was achieved
only in the third decade of the 20th century.
More precisely, William Rowan Hamilton put
forth the quaternions and biquaternions; James
Cockle presented tessarines and coquaternions;
and William Kingdon Clifford was an enthusiast
of split-biquaternions, which he called algebraic
motors. These noncommutative algebras, and
the non-associative Lie algebras, were studied
within universal algebra before the subject
was divided into particular mathematical structure
types. One sign of re-organization was the
use of direct sums to describe algebraic structure.
The various hypercomplex numbers were identified
with matrix rings by Joseph Wedderburn (1908)
and Emil Artin (1928). Wedderburn's structure
theorems were formulated for finite-dimensional
algebras over a field while Artin generalized
them to Artinian rings.
In 1920, Emmy Noether, in collaboration with
W. Schmeidler, published a paper about the
theory of ideals in which they defined left
and right ideals in a ring. The following
year she published a landmark paper called
Idealtheorie in Ringbereichen, analyzing ascending
chain conditions with regard to (mathematical)
ideals. Noted algebraist Irving Kaplansky
called this work "revolutionary"; the publication
gave rise to the term "Noetherian ring", and
several other mathematical objects being called
Noetherian.
== Notes
