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Ring (mathematics)
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This article is about algebraic structures. For other uses, see Ring.


An image illustrating geometric addition on a cubic curve in projective
space. Ring theory has many important applications in algebraic geometry.
In mathematics, a ring is an algebraic structure consisting of a set
together with two binary operations (usually called addition and
multiplication), where each operation combines two elements to form a third
element. To qualify as a ring, the set together with its two operations must
satisfy certain conditions-namely, the set must be an abelian group under
addition and a monoid under multiplicationa[>] such that multiplication
distributes over addition. While these operations are familiar from many
mathematical structures, such as number systems or the integers-for example,
they are also very general in the sense that they take a broad variety of
mathematical objects into account. This allows one to handle entities of
very different mathematical origins in a flexible way, while retaining
essential structural aspects of many objects in abstract algebra and beyond.
The ubiquity of rings makes them a central organizing principle of
contemporary mathematics. The branch of mathematics that studies rings is
known as ring theory.[1]
Rings share a fundamental kinship with number theory and linear algebra. The
former is responsible for various analogues of number-theoretic theorems in
ring theory-for instance, the fundamental theorem of arithmetic translates
to a certain special class of rings known as unique factorization domains.
The latter is responsible for the rich theory of algebras, including, but
not restricted to, matrix rings. This theory of matrix rings, for example,
is a striking consequence of the ways in which noncommutative ring theory
may be used to understand fundamental physical laws underlying special
relativity and symmetry phenomena in molecular chemistry.
The concept of a ring first arose from attempts to prove Fermat's last
theorem, starting with Richard Dedekind in the 1880's. After contributions
from other fields, mainly number theory, the ring notion was generalized and
firmly established during the 1920's by Emmy Noether and Wolfgang Krull.[2]
Modern ring theory-a very active mathematical discipline-studies rings in
their own right. To explore rings, mathematicians have devised various
notions to break rings into smaller, better-understandable pieces, such as
ideals, quotient rings and simple rings. In addition to these abstract
properties, ring theorists also make various distinctions between the theory
of commutative rings and noncommutative rings-the former belonging to
algebraic number theory and algebraic geometry. A particularly rich theory
has been developed for a certain special class of commutative rings, known
as fields, which lies within the realm of field theory. Likewise, the
corresponding theory for noncommutative rings, that of noncommutative
division rings, constitutes an active research interest for noncommutative
ring theorists. Since the discovery of a mysterious connection between
noncommutative ring theory and geometry during the 1980's by Alain Connes,
noncommutative geometry has become a particularly active discipline in ring
theory.
Contents
[hide]
1 Definition and illustration
1.1 First example: the integers
1.2 Formal definition
1.3 Notes on the definition
1.4 Second example: the ring Z4
1.5 Third example: the trivial ring
2 Basic concepts
2.1 Subring
2.2 Homomorphism
2.3 Ideal
3 History
4 Elementary properties of rings
5 New rings from old
5.1 Quotient ring
5.2 Polynomial ring
5.3 Matrix ring
6 Some examples of the ubiquity of rings
7 Special rings and rings with additional structure
7.1 Finite ring
7.2 Associative algebras
7.3 Lie ring
7.4 Topological ring
8 Commutative rings
8.1 Principal ideal domains
8.2 Unique factorization domains
8.3 Integral domains and fields
8.3.1 Relation to other algebraic structures
9 Noncommutative rings
10 Category theoretical description
11 See also
12 Notes
12.1 Citations
13 References
13.1 General references
13.2 Special references
13.3 Historical references


[edit] Definition and illustration
[edit] First example: the integers
v . d . eAlgebraic structures

MagmaSet S with binary operation +

SemigroupAssociativity of +

MonoidExistence of identity element for + in S

GroupExistence of inverse elements for + in S

Abelian groupCommutativity of +

Pseudo-ringAssociative binary operation �
Distributivity of � over +

RingExistence of identity element for � in S

Commutative ringCommutativity of �

FieldExistence of inverse elements for � in S

The most familiar example of a ring is the set of all integers, Z,
consisting of the numbers
...., ?4, ?3, ?2, ?1, 0, 1, 2, 3, 4, ... [3]
together with the usual operations of addition and multiplication. These
operations satisfy the following properties:
The integers form an abelian group under addition; that is:
Closure axiom for addition: Given two integers a and b, their sum, a + b is
also an integer.
Existence of additive identity: For any integer a, a + 0 = 0 + a = a. Zero
is called the identity element of the integers because adding 0 to any
integer (in any order) returns the same integer.
Commutativity of addition: For any two integers a and b, a + b = b + a. So
the order in which two integers are added is irrelevant.
Associativity of addition: For any integers, a, b and c, (a + b) + c = a +
(b + c). So, adding b to a, and then adding c to this result, is the same as
adding c to b, and then adding this result to a.
Existence of additive inverse: For any integer a, there exists an integer
denoted by ?a such that a + (?a) = (?a) + a = 0. The element, ?a, is called
the additive inverse of a because adding a to ?a (in any order) returns the
identity.
The integers form a multiplicative monoid (a monoid under multiplication);
that is:
Closure axiom for multiplication: Given two integers a and b, their product,
a � b is also an integer.
Associativity of multiplication: Given any integers, a, b and c, (a � b) � c
= a � (b � c). So multiplying b with a, and then multiplying c to this
result, is the same as multiplying c with b, and then multiplying a to this
result.
Existence of multiplicative identity: For any integer a, a � 1 = 1 � a = a.
So multiplying any integer with 1 (in any order) gives back that integer.
One is therefore called the multiplicative identity.
Multiplication is distributive over addition : These two structures on the
integers (addition and multiplication) are compatible in the sense that
a � (b + c) = (a � b) + (a � c), and
(a + b) � c = (a � c) + (b � c)
for any three integers a, b and c.
[edit] Formal definition
There are some differences in exactly what axioms are used to define a ring.
Here one set of axioms is given, and comments on variations follow.
A ring is a set R equipped with two binary operations + : R � R ? R and � :
R � R ? R (where � denotes the Cartesian product), called addition and
multiplication. To qualify as a ring, the set and two operations, (R, +,
� ), must satisfy the following requirements known as the ring axioms. [4]

(R, +, � ) is required to be an abelian group under addition:
1.Closure under addition.For all a, b in R, the result of the operation a +
b is also in R.c[>]
2.Associativity of addition.For all a, b and c in R, the equation (a + b) +
c = a + (b + c) holds.
3.Existence of additive identity.There exists an element 0 in R, such that
for all elements a in R, the equation 0 + a = a + 0 = a holds.
4.Existence of additive inverse.For each a in R, there exists an element b
in R such that a + b = b + a = 0
5.Commutativity of addition.For all a, b in R, the equation a + b = b + a
holds.

(R, +, � ) is required to be a monoid under multiplication:
1.Closure under multiplication.For all a, b in R, the result of the
operation a � b is also in R.c[>]
2.Associativity of multiplication.For all a, b, and c in R, the equation (a
� b) � c = a � (b � c) holds.
3.Existence of multiplicative identity.a[>]There exists an element 1 in R,
such that for all elements a in R, the equation 1 � a = a � 1 = a holds.

The distributive laws:
1. For all a, b and c in R, the equation a � (b + c) = (a � b) + (a � c)
holds.
2. For all a, b and c in R, the equation (a + b) � c = (a � c) + (b � c)
holds.


This definition assumes that a binary operation on R is a function defined
on R�R with values in R. Therefore, for any a and b in R, the addition a + b
and the product a � b are elements of R.
The most familiar example of a ring is the set of all integers, Z = {...,
?4, ?3, ?2, ?1, 0, 1, 2, 3, 4, ... }, together with the usual operations of
addition and multiplication.[5]
Another familiar example is the set of real numbers R, equipped with the
usual addition and multiplication.
Another example of ring is the set of all square matrices of a fixed size,
with real elements, using the matrix addition and multiplication of linear
algebra. In this case, the ring elements 0 and 1 are the zero matrix (with
all entries equal to 0) and the identity matrix, respectively.
[edit] Notes on the definition
As with some axiomatic theories, there are often differences of usage in
what axioms a ring should satisfy. Sometimes the disagreement between two
definitions is minor. For instance, some authors insist that 1 ? 0 in a ring
(in words, this means that the multiplicative identity of the ring must be
different from its additive identity). In particular they do not consider
the trivial ring to be a ring (see below).
A more significant disagreement is that some authors omit the existence of a
multiplicative identity in a ring[6][7] [8] . For instance, this would allow
the even integers to form a ring with the natural operations of addition and
multiplication (all ring axioms are satisfied except for the existence of a
multiplicative identity). Rings that satisfy the ring axioms as given above
but do not contain a multiplicative identity are sometimes called
pseudo-rings. The term rng (jocular; ring without the multiplicative
identity) is also used for such rings. Rings which do have multiplicative
identities (and also satisfy the above axioms) are sometimes referred to
unital rings, unitary rings, rings with unity, rings with identity or rings
with 1.[9] Note that one can always embed a non-unitary ring inside a
unitary ring (see this for one particular construction of this embedding).
There are still other more significant differences between two particular
definitions of a ring. For instance, some authors omit associativity of
multiplication in the set of ring axioms; rings that are nonassociative are
called nonassociative rings. In this article, all rings are assumed to
satisfy the axioms as given above unless stated otherwise.
[edit] Second example: the ring Z4
Consider the set Z4 consisting of the numbers 0, 1, 2, 3 where addition and
multiplication are defined as follows (note that for any integer, x, x mod 4
is defined to be the remainder when x is divided by 4):
For any x, y in Z4, x + y is defined to be their sum in Z (the set of all
integers) mod 4. So we can represent the additive structure of Z4 by the
left-most table as shown.
For any x, y in Z4, x ? y is defined to be their product in Z (the set of
all integers) mod 4. So we can represent the multiplicative structure of Z4
by the right-most table as shown.
�0123
00000
10123
20202
30321
+0123
00123
11230
22301
33012
It is simple (but tedious) to verify that Z4 is a ring under these
operations. First of all, one can use the left-most table to show that Z4 is
closed under addition (any result is either 0, 1, 2 or 3). Associativity of
addition in Z4 follows from associativity of addition in the set of all
integers. The additive identity is 0 as can be verified by looking at the
left-most table. Given an integer x, there is always an inverse of x; this
inverse is given by 4 - x as one can verify from the additive table.
Therefore, Z4 is an abelian group under addition.
Similarly, Z4 is closed under multiplication as the right-most table shows
(any result above is either 0, 1, 2 or 3). Associativity of multiplication
in Z4 follows from associativity of multiplication in the set of all
integers. The multiplicative identity is 1 as can be verified by looking at
the right-most table. Therefore, Z4 is a monoid under multiplication.
Distributivity of the two operations over each other follow from
distributivity of addition over multiplication (and vice-versa) in Z (the
set of all integers).
Therefore, this set does indeed form a ring under the given operations of
addition and multiplication.
Properties of this ring
In general, given any two integers, x and y, if x ? y = 0, then either x is
0 or y is 0. It is interesting to note that this does not hold for the ring
(Z4, +, ?):
2 ? 2 = 0
although neither factor is 0. In general, a non-zero element a of a ring,
(R, +, ?) is said to be a zero divisor in (R, +, ?), if there exists a
non-zero element b of R such that a ? b = 0. So in this ring, the only zero
divisor is 2 (note that 0 ? a = 0 for any a in a ring (R, +, ?) so 0 is not
considered to be a zero divisor).
A commutative ring which has no zero divisors is called an integral domain
(see below). So Z, the ring of all integers (see above), is an integral
domain (and therefore a ring), although Z4 (the above example) does not form
an integral domain (but is still a ring). So in general, every integral
domain is a ring but not every ring is an integral domain.
[edit] Third example: the trivial ring
If we define on the singleton set {0}:
0 + 0 = 0
0 � 0 = 0
then one can verify that ({0}, +, �) forms a ring known as the trivial ring.
Since there can be only one result for any product or sum (0), this ring is
both closed and associative for addition and multiplication, and furthermore
satisfies the distributive law. The additive and multiplicative identities
are both equal to 0. Similarly, the additive inverse of 0 is 0. The trivial
ring is also a (rather trivial) example of a zero ring (see below).
[edit] Basic concepts
[edit] Subring
Informally, a subring of a ring is another ring that uses the "same"
operations and is contained in it. More formally, suppose (R, +, �) is a
ring, and S is a subset of R such that
for every a, b in S, a + b is in S;
for every a, b in S, a � b is in S;
for every a in S, the additive inverse ?a of a is in S; and
the multiplicative identity '1' of R is in S.
Let '+S' and '�S' denote the operations '+' and '�', restricted to S�S. Then
(S, +S, �S) is a subring of (R, +, �).[10] Since the restricted operations
are completely determined by S and the original ones, the subring is often
written simply as (S, +, �).
For example, a subring of the complex number ring C is any subset of C that
includes 1 and is closed under addition, multiplication, and negation, such
as:
The rational numbers Q
The algebraic numbers A
The real numbers R
If A is a subring of R, and B is a subset of A such that B is also a subring
of R, then B is a subring of A.
[edit] Homomorphism
A homomorphism from a ring (R, +, �) to a ring (S, ?, *) is a function f
from R to S that commutes with the ring operations; namely, such that, for
all a, b in R the following identities hold:
f(a + b) = f(a) ? f(b)
f(a � b) = f(a) * f(b)
Moreover, the function f must take the identity element 1R of '�' to the
identity element 1S of '*'.
For example, the function that maps each integer x to its remainder modulo 4
(a number in {0, 1, 2, 3}) is a homomorphism from the ring Z to the ring Z4.
If f is a ring homomorphism from (R, +, �) to (S, ?, *), the inverse image
of the identity element 1S of ? (that is, all elements of R that are mapped
to 1S by f) is a subring of (R, +, �).
A ring homomorphism is said to be an isomorphism if it is both an
epimorphism and a monomorphism in the category of rings.
[edit] Ideal
Main article: Ideal (ring theory)
The purpose of an ideal in a ring is to somehow allow one to define the
quotient ring of a ring (analogous to the quotient group of a group; see
below). An ideal in a ring can therefore be thought of as a generalization
of a normal subgroup in a group. More formally, let (R, +, � ) be a ring. A
subset I of R is said to be a right ideal in R if:
(I, +) is a subgroup of the underlying additive group in (R, +, � ) (i.e (I,
+) is a subgroup of (R, +)).
For every x in I and r in R, x � r is in I.
A left ideal is similarly defined with the second condition being replaced.
More specifically, a subset I of R is a left ideal in R if:
(I, +) is a subgroup of the underlying additive group in (R, +, � ) (i.e (I,
+) is a subgroup of (R, +)).
For every x in I and r in R, r � x is in I.
Notes
If k is in R, then k � R is a right ideal in R, and R � k is a left ideal in
R. These ideals (for any k in R) are called the principal right and left
ideals generated by k.
If every ideal in a ring (R, +, � ) is a principal ideal in (R, +, � ), (R,
+, � ) is said to be a principal ideal ring.
An ideal in a ring, (R, +, � ), is said to be a two-sided ideal if it is
both a left ideal and right ideal in (R, +, � ). It is preferred to call a
two-sided ideal, simply an ideal.
If I = {0} (where 0 is the additive identity of the ring (R, +, � )), then I
is an ideal known as the trivial ideal. Similarly, R is also an ideal in (R,
+, � ) called the unit ideal.
Examples
Any additive subgroup of the integers is an ideal in the integers with its
natural ring structure.
There are no non-trivial ideals in R (the ring of all real numbers) (i.e,
the only ideals in R are {0} and R itself). More generally, a field cannot
contain any non-trivial ideals.
From the previous example, every field must be a principal ideal ring.
A subset, I, of a commutative ring (R, +, � ) is a left ideal if and only if
it is a right ideal. So for simplicity's sake, we refer to any ideal in a
commutative ring as just an ideal.
[edit] History
Main article: Ring theory#History


A portrait of Richard Dedekind: the founder of ring theory.
The study of rings originated from the theory of polynomial rings and the
theory of algebraic integers. Furthermore, the appearance of hypercomplex
numbers in the mid-nineteenth century undercut the pre-eminence of fields in
mathematical analysis.
Richard Dedekind (image to the right) introduced the concept of a ring, [2]
and the term ring (Zahlring) was coined by David Hilbert in 1892 and
published in the article Die Theorie der algebraischen Zahlk�rper,
Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. 4, 1897.
According to Harvey Cohn, Hilbert used the term for a specific ring that had
the property of "circling directly back" to an element of itself.[11]
The first axiomatic definition of a ring was given by Adolf Fraenkel in an
essay in Journal f�r die reine und angewandte Mathematik (A. L. Crelle),
vol. 145, 1914. [2] In 1921, Emmy Noether gave the first axiomatic
foundation of the theory of commutative rings in her monumental paper Ideal
Theory in Rings.[2]
[edit] Elementary properties of rings
Some properties of rings follow directly from the ring axioms through very
simple proofs.
In particular, since the axioms state that (R,+) is a commutative group, all
pertinent theorems of group theory apply, such as the uniqueness of the
additive identity and the uniqueness of the additive inverse of a particular
element. In the same way one can prove the uniqueness of the inverse of a
unit in a ring.
However, rings also have specific properties that combine addition with
multiplication. In any ring (R, +, ?):
For any element a, the equations a ? 0 = 0 ? a = 0 hold.
If 0 = 1, the ring is trivial (that is, R has only one element).
For any element a, the equation (?1) ? a = ?a holds.
For any elements a and b, the equations (?a) ? b = a ? (?b) = ?(a ? b)
[edit] New rings from old
[edit] Quotient ring
Main article: Quotient ring
Informally, the quotient ring of a ring, is a generalization of the notion
of a quotient group of a group. More formally, given a ring (R, +, � ) and a
two-sided ideal I of (R, +, � ), the quotient ring (or factor ring) R/I is
the set of cosets of I (with respect to the underlying additive group of (R,
+, � ); i.e cosets with respect to (R, +)) together with the operations:
(a + I) + (b + I) = (a + b) + I and
(a + I)(b + I) = (ab) + I.
For every a, b in R.
[edit] Polynomial ring
Main article: Polynomial ring
Formal definition
Let (R, +R, �R) be a ring and let where the convention that is the set of
all nonnegative integers, is adopted. Define +S : S � S ? S and �S : S � S ?
S, as follows (where and are arbitrary elements of S):


Then (S, +S, �S) is a ring referred to as the polynomial ring over R.
Notes
Most authors write S as R[X], where is an element of S. This convention is
adopted because if is an element of S, and if , f can be written as a
polynomial with coefficients in R, in the variable X, as follows:

This allows one to view S as merely the set of all polynomials over R in the
variable X, with multiplication and addition of polynomials defined in the
canonical manner. Therefore, in all that follows, S, denoted by R[X], shall
be identified in this fashion. Addition and multiplication in S, and those
of the underlying ring R, will be denoted by juxtaposition.
[edit] Matrix ring
Main article: Matrix ring
Formal definition
Let (R, +R, �R) be a ring and let . Define +M : Mr(R) � Mr(R) ? Mr(R) and �M
: Mr(R) � Mr(R) ? Mr(R), as follows (where and are arbitrary elements of
Mr(R)):


Then (Mr(R), +M, �M) is a ring referred to as the ring of r�r matrices over
R.
Notes
The ring Mr(R) can be viewed as merely the ring of r�r matrices over R with
matrix addition and multiplication defined in the canonical manner. For
given in Mr(R), f can be identified with the r�r matrix whose (i, j)-entry
is fij. Therefore, in all that follows, Mr(R) and each of its elements shall
be identified in this fashion. Addition and multiplication in Mr(R), and
those of the underlying ring R, will be denoted by juxtaposition.
[edit] Some examples of the ubiquity of rings
It is remarkable how many different kinds of mathematical objects can be
fruitfully analyzed in terms of some associated ring. For instance:
To any topological space X one can associate its integral cohomology ring ,
a graded ring. As the name suggests, there are also homology groups of a
space, and indeed these were defined first, as a useful tool for
distinguishing between certain pairs of topological spaces, like the spheres
and tori, for which the methods of point-set topology are not well-suited.
Cohomology groups were later defined in terms of homology groups in a way
which is roughly analogous to the dual of a vector space. To know each
individual integral homology group is essentially the same as knowing each
individual integral cohomology group, because of the universal coefficient
theorem. However, the advantage of the cohomology groups is that there is a
natural product, which is analogous to the observation that one can multiply
pointwise a k-multilinear form and an l-multilinear form to get a (k +
l)-mulilinear form. The importance of the ring structure in cohomology is
hard to overstate: it provides the foundation for characteristic classes of
fiber bundles, intersection theory on manifolds and algebraic varieties,
Schubert calculus and much more.
To any group is associated its Burnside ring which uses a ring to describe
the various ways the group can act on a finite set. The Burnside ring's
additive group is the free abelian group whose basis are the transitive
actions of the group and whose addition is the disjoint union of the action.
Expressing an action in terms of the basis is decomposing an action into its
transitive constituents. The multiplication is easily expressed in terms of
the representation ring: the multiplication in the Burnside ring is formed
by writing the tensor product of two permutation modules as a permutation
module. The ring structure allows a formal way of subtracting one action
from another. Since the Burnside ring is contained as a finite index subring
of the representation ring, one can pass easily from one to the other by
extending the coefficients from integers to the rational numbers.
To any group ring or Hopf algebra is associated its representation ring or
"Green ring". The representation ring's additive group is the free abelian
group whose basis are the indecomposable modules and whose addition
corresponds to the direct sum. Expressing a module in terms of the basis is
finding an indecomposable decomposition of the module. The multiplication is
the tensor product. When the algebra is semisimple, the representation ring
is just the character ring from character theory, which is more or less the
Grothendieck group given a ring structure.
To any irreducible algebraic variety is associated its function field. The
points of an algebraic variety correspond to valuation rings contained in
the function field and containing the coordinate ring. The study of
algebraic geometry makes heavy use of commutative algebra to study geometric
concepts in terms of ring-theoretic properties. Birational geometry studies
maps between the subrings of the function field.
Every simplicial complex has an associated face ring, also called its
Stanley-Reisner ring. This ring reflects many of the combinatorial
properties of the simplicial complex, so it is of particular interest in
algebraic combinatorics. In particular, the algebraic geometry of the
Stanley-Reisner ring was used to characterize the numbers of faces in each
dimension of simplicial polytopes.
[edit] Special rings and rings with additional structure
[edit] Finite ring
Main article: Finite ring
Given a natural number m, how many distinct rings (not necessarily with
unity) have m elements? When m is prime there are only two rings of order m;
the additive group of each being isomorphic to the cyclic group of order m.
One is the zero ring and the other is the Galois field.
As with finite groups, the complexity of the classification depends upon the
complexity of the prime factorization of m. If m is the square of a prime,
for instance, there are precisely eleven rings having order m. On the other
hand, there can be only two groups having order m; both of which are
abelian.
The theory of finite rings is more complex than that of finite abelian
groups, since any finite abelian group is the additive group of at least two
nonisomorphic finite rings: the direct product of copies of , and the zero
ring. On the other hand, the theory of finite rings is simpler than that of
not necessarily abelian finite groups. For instance, the classification of
finite simple groups was one of the major breakthroughs of twentieth century
mathematics, its proof spanning thousands of journal pages. On the other
hand, any finite simple ring is isomorphic to the ring of n by n matrices
over a finite field of order q. This follows from two theorems of Joseph
Wedderburn established in 1905 and 1907.
One of these theorems, known as Wedderburn's little theorem, asserts that
any finite division ring is necessarily commutative. Nathan Jacobson later
discovered yet another condition which guarantees commutativity of a ring:
If for every element r of R there exists an integer n > 1 such that rn = r,
then R is commutative.[12]
If, r2 = r for every r, the ring is called a Boolean ring. More general
conditions which guarantee commutativity of a ring are also known.[13]
The number of rings with m elements, for m a natual number, is listed under
A027623 in the On-Line Encyclopedia of Integer Sequences.
[edit] Associative algebras
An associative algebra is a ring that is also a vector space over a field K.
For instance, the set of n by n matrices over the real field R has dimension
n2 as a real vector space, and the matrix multiplication corresponds to the
ring multiplication. For a non-trivial but elementary example consider 2 � 2
real matrices.
[edit] Lie ring
A Lie ring is defined to be a ring that is nonassociative and
anticommutative under multiplication, that also satisfies the Jacobi
identity. More specifically we can define a Lie ring L to be an abelian
group under addition with an operation that has the following properties:
Bilinearity:

for all x, y, z ? L.
Jacobi identity:

for all x, y, z in L.
For all x in L.

Lie rings need not be Lie groups under addition. Any Lie algebra is an
example of a Lie ring. Any associative ring can be made into a Lie ring by
defining a bracket operator [x,y] = xy ? yx. Conversely to any Lie algebra
there is a corresponding ring, called the universal enveloping algebra.
Lie rings are used in the study of finite p-groups through the Lazard
correspondence. The lower central factors of a p-group are finite abelian
p-groups, so modules over Z/pZ. The direct sum of the lower central factors
is given the structure of a Lie ring by defining the bracket to be the
commutator of two coset representatives. The Lie ring structure is enriched
with another module homomorphism, then pth power map, making the associated
Lie ring a so-called restricted Lie ring.
Lie rings are also useful in the definition of a p-adic analytic groups and
their endomorphisms by studying Lie algebras over rings of integers such as
the p-adic integers. The definition of finite groups of Lie type due to
Chevalley involves restricting from a Lie algebra over the complex numbers
to a Lie algebra over the integers, and the reducing modulo p to get a Lie
algebra over a finite field.
[edit] Topological ring
Let (X, T) is a topological space and (X, +, � ) be a ring. Then (X, T, +,
� ) is said to be a topological ring, if its ring structure and topological
structure are both compatible (i.e work together) over each other. That is,
the addition map ( ) and the multiplication map ( ) have to be both
continuous as maps between topological spaces where X x X inherits the
product topology. So clearly, any topological ring is a topological group
(under addition).
Examples
The set of all real numbers, R, with its natural ring structure and the
standard topology forms a topological ring.
The direct product of two topological rings is also a topological ring.
[edit] Commutative rings
Main article: Commutative ring
Although ring addition is commutative, so that for every a, b in R, a + b =
b + a, ring multiplication is not required to be commutative; a � b need not
equal b � a for all a, b in R. Rings that also satisfy commutativity for
multiplication are called commutative rings.e[>] Formally,
Formal definition
Let (R, +, � ) be a ring. Then (R, +, � ) is said to be a commutative ring
if for every a, b in R, a � b = b � a. That is, (R, +, � ) is required to be
a commutative monoid under multiplication.
Examples
The integers form a commutative ring under the natural operations of
addition and multiplication.
An example of a noncommutative ring is the ring of n � n matrices over a
non-trivial field K, for n > 1. In particular, the ring of all 2 � 2
matrices over R (the set of all real numbers) do not form a commutative ring
as the following computation shows:
, which is not equal to
[edit] Principal ideal domains
Main article: Principal ideal domain
Although rings are structurally similar to the integers, there are certain
ring-theoretic properties that the integers may satisfy but a general ring
may not. One such property is the requirement that every ideal in a ring be
generated by a single element; that is, be a principal ideal. Formally,
Definition
Let R be a ring. Then R is said to be a principal ideal ring (abbreviated
PIR), if every ideal in R is of the form a � R = {a � r | r � R}. A
principal ideal domain is a principal ideal ring that is also an integral
domain.
The requirement that a ring be a principal ideal domain is somewhat stronger
than the other more common properties a ring may satisfy. For example, it is
true that if a ring, R, is a unique factorization domain (UFD), then the
polynomial ring over R is also a UFD. However, such a result does not in
general hold for principal ideal rings. For example, the integers are an
easy example of a principal ideal ring, but the polynomial ring over the
integers fails to be a PIR; if R = Z[x] denotes the polynomial ring over the
integers, I = 2 � R + X � R is an ideal which cannot be generated by a
single element. Despite this counterexample, the polynomial ring over any
field is always a principal ideal domain and in fact, a Euclidean domain.
More generally, a polynomial ring is a PID if and only if the polynomial
ring in question is over a field.
Aside from the polynomial ring over a PIR, principal ideal rings possess
many interesting properties due to their connection with the integers in
terms of divisibility; that is, principal ideal domains behave similarly to
the integers with respect to divisibility. For example, any PID is a UFD;
i.e, an analogue of the fundamental theorem of arithmetic holds for
principal ideal domains. Furthermore, since Noetherian rings are precisely
those rings in which any ideal is finitely generated, principal ideal
domains are trivially Noetherian rings. The fact that irreducible elements
coincide with prime elements for PID's, together with the fact that every
PID is Noetherian, implies that any PID is a UFD. One can also speak of the
greatest common divisor of two elements in a PID; if x and y are elements of
R, a principal ideal domain, then x � R + y � R = c � R for some c in R
since the left-hand side is indeed an ideal. Therefore, c is the desired
"GCD" of x and y.
An important class of rings lying in between fields and PID's, is the class
of Euclidean domains. In particular, any field is a Euclidean domain, and
any Euclidean domain is a PID. An ideal in a Euclidean domain is generated
by any element of that ideal with minimum degree (all such elements must be
associate). However, not every PID is a Euclidean domain; the ring
furnishes a counterexample.
[edit] Unique factorization domains
Main article: Unique factorization domain
The theory of unique factorization domains (UFD) also forms an important
part of ring theory. In effect, a unique factorization domain is ring in
which an analogue of the fundamental theorem of arithmetic holds. Formally,
Definition
Let R be a ring. Then R is said to be a unique factorization domain
(abbreviated UFD), if the following conditions are satisfied:
1. R is an integral domain.
2. Every non-zero non-unit of R is the product of a finite number of
irreducible elements.
3. If = , where all ai's and bj's are irreducible, then n = m and after
possible renumbering of the ai's and bj's, bi = ai � ui where ui is a unit
in R.
The second condition above guarantees that "non-trivial" elements of R can
be decomposed into irreducibles, and according to the third condition, such
a finite decomposition is unique "up to multiplication by unit elements."
This weakened form of uniqueness is reasonable to assume for otherwise even
the integers would not satisfy the properties of being a UFD ((-2)2 = 22 = 4
demonstrates two "distint" decompositions of 4; however both decompositions
of 4 are equivalent up to multiplication by units (-1 and +1)). The fact
that the integers constitute a UFD follows from the fundamental theorem of
arithmetic.
For arbitrary rings, one may define a prime element and an irreducible
element; these two may not in general coincide. However, a prime element in
a domain is always irreducible. For UFD's, irreducible elements are also
primes.
The class of unique factorization domains is related to other classes of
rings. For instance, any Noetherian domain satisfies conditions 1 and 2
above, but in general Noetherian domains fail to satisfy condition 3.
However, if the set of prime elements and the set of irreducible elements
coincide for a Noetherian domain, the third condition of a UFD is satisfied.
In particular, principal ideal domains are UFD's.
[edit] Integral domains and fields
Main articles: Integral domain and field
While rings are very important mathematical objects, there are a lot of
restrictions involved in their theory. For instance, suppose we have a ring
R and suppose a and b are in R. If a is non-zero and a � b = 0, then b need
not necessarily be 0. In particular, if a � b = a � c with a non-zero b need
not equal c. An example of this is the set of n x n matrices over R where a
maybe such a non-zero matrix but may be singular. In this case, the result
may not be true. However, we can impose additional conditions on the ring to
ensure that this be true; namely make the ring into an integral domain (that
is a non-trivial commutative ring with no zero divisors). But we still run
into a problem; namely we can't necessarily divide by non-zero elements. For
example, the collection of all integers form an integral domain but we still
can't divide an integer a by another integer b. For example, 2 cannot divide
3 to obtain another element in this ring. However, this problem can readily
be solved if we ensure that every element in the ring has a multiplicative
inverse. A field is a ring in which the non-zero elements form an abelian
group under multiplication. In particular, a field is an integral domain
(and therefore has no zero divisors) along with an additional operation of
'division'. Namely, if a and b are in a field F, then a/b is defined to be a
� b-1 which is well defined.
Formal definition
Let (R, +, � ) be a ring. Then (R, +, � ) is said to be an integral domain
if (R, +, � ) is commutative and has no zero divisors. Furthermore, (R, +,
� ) is said to be a field, if its non-zero elements form an abelian group
under multiplication.
Note
If we require that the zero element (0) in a ring to have a multiplicative
inverse, then the ring must be trivial.
Examples
The integers form a commutative ring under the natural operations of
addition and multiplication. In fact, the integers form what is known as an
integral domain (a commutative ring with no zero divisors).
A ring whose non-zero elements form an abelian group under multiplication
(not just a commutative monoid), is called a field. So every field is an
integral domain and every integral domain is a commutative ring.
Furthermore, any finite integral domain is a field.
[edit] Relation to other algebraic structures
The following is a chain of class inclusions that describes the relationship
between rings, domains and fields:
Commutative rings ? integral domains ? half factorization domains ?unique
factorization domains ? principal ideal domains ? Euclidean domains ? fields
Fields and integral domains are very important in modern algebra.
[edit] Noncommutative rings
Main article: Ring theory
The study of noncommutative rings is a major area in modern algebra;
especially ring theory. Often noncommutative rings possess interesting
invariants that commutative rings do not. As an example, there exist rings
which contain non-trivial proper left or right ideals, but are still simple;
that is contain no non-trivial proper (two-sided) ideals. This example
illustrates how one must take care when studying noncommutative rings
because of possible counterintuitive misconceptions.
The theory of vector spaces is one illustration of a special case of an
object studied in noncommutative ring theory. In linear algebra, the
"scalars of a vector space" are required to lie in a field - a commutative
division ring. The concept of a module, however, requires only that the
scalars lie in an abstract ring. Neither commutativity nor the division ring
assumption is required on the scalars in this case. Module theory has
various applications in noncommutative ring theory, as one can often obtain
information about the structure of a ring by making use of its modules. 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
remarkable 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. Therefore, the
Jacobson radical also captures a concept which may seem to be not
well-defined for noncommutative rings.
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 physics. More
generally, endomorphism rings of abelian groups are rarely commutative.
Noncommutative rings, like noncommutative groups, are not very well
understood. For instance, although every finite abelian group is the direct
sum of (finite) cyclic groups of prime-power order, non-abelian groups do
not possess such a simple structure. Likewise, various invariants exist for
commutative rings, whereas invariants of noncommutative rings are difficult
to find. As an example, the nilradical, although "innocent" in nature, need
not be an ideal unless the ring is assumed to be 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. Therefore, the nilradical cannot be studied in noncommutative ring
theory. Note however that there are analogues of the nilradical defined for
noncommutative rings, that coincide with the nilradical when commutativity
is assumed.
One of the best known noncommutative rings is the division ring of
quaternions.
[edit] Category theoretical description
Every ring can be thought of as monoids in Ab, the category of abelian
groups (thought of as a monoidal category under the tensor product). The
monoid action of a ring R on a abelian group is simply an R-module.
Essentially, an R-module is a generalization of the notion of a vector
space - where rather than a vector space over a field, one has a "vector
space over a ring".
Let (A, +) be an abelian group and let End(A) be its endomorphism ring (see
above). Note that, essentially, End(A) is the set of all morphisms of A,
where if f is in End(A), and g is in End(A), the following rules may be used
to compute f + g and f � g:
(f + g)(x) = f(x) + g(x)
(f � g)(x) = f(g(x))
where + as in f(x) + g(x) is addition in A, and function composition is
denoted from right to left. Therefore, associated to any abelian group, is a
ring. Conversly, given any ring, (R, +, � ), (R, +) is an abelian group.
Furthermore, for every r in R, right (or left) multiplication by r gives
rise to a morphism of (R, +), by right (or left) distributivity. Let A = (R,
+). Consider those endomorphisms of A, that "factor through" right (or left)
multiplication of R. In other words, let EndR(A) be the set of all morphisms
m of A, having the property that m(r � x) = r � m(x). It was seen that every
r in R gives rise to a morphism of A - right multiplication by r. It is in
fact true that this association of any element of R, to a morphism of A, as
a function from R to EndR(A), is an isomorphism of rings. In this sense,
therefore, any ring can be viewed as the endomorphism ring of some abelian
X-group (by X-group, it is meant a group with X being its set of operators).
In essence, the most general form of a ring, is the endomorphism group of
some abelian X-group.
[edit] See also
Wikibooks has a book on the topic of
Abstract algebra/Rings, fields and modules

Ring theory
Glossary of ring theory
Category of rings
Algebra over a commutative ring
Nonassociative ring
Algebraic structure
Chinese remainder theorem
Semiring
Special types of rings:
Boolean ring
Commutative ring
Ordered ring
Noetherian and artinian rings
Dedekind ring
Differential ring
Division ring (skew field)
Exponential ring
Field
Integral domain (ID)
Local ring
Principal ideal domain (PID)
Reduced ring
Regular ring
Unique factorization domain (UFD)
Valuation ring and discrete valuation ring
Zero ring
[edit] Notes
^ a: Some authors only require that a ring be a semigroup under
multiplication; that is, do not require that there be a multiplicative
identity (1). See the section Notes on the definition for more details.
^ b: Elements which do have multiplicative inverses are called units, see
Lang 2002, �II.1, p. 84 or this for a treatment of units in this article.
^ c: The closure axiom is already implied by the condition that +/. be a
binary operation. Some authors therefore omit this axiom. Lang 2002
^ d: The transition from the integers to the rationals by adding fractions
is generalized by the quotient field.
^ e: Many authors include commutativity of rings in the set of ring axioms
(see above) and therefore refer to "commutative rings" as just "rings".

[edit] Citations
^ Herstein 1964, �3, p. 83
^ a b c d [1]
^ Lang 2005, App. 2, p. 360
^ Herstein 1975, �2.1, p. 27
^ Lang 2005, App. 2, p. 360
^ Herstein, I. N. Topics in Algebra, Wiley; 2 edition (June 20, 1975), ISBN
0-471-01090-1.
^ Joseph Gallian (2004). Contemporary Abstract Algebra. Houghton Mifflin.
ISBN 9780618514717.
^ Neal H. McCoy (1964). The Theory of Rings. The MacMillian Company. pp.
161. ISBN 978-1124045559.
^ Raymond Louis Wilder (1965). Introduction to Foundations of Mathematics.
John Wiley and Sons. p. 176.
^ Lang 2005, �II.1, p. 90
^ Cohn, Harvey (1980). Advanced Number Theory. New York: Dover Publications.
pp. 49. ISBN 9780486640235.
^ Jacobson 1945
^ Pinter-Lucke 2007
[edit] References
[edit] General references
R.B.J.T. Allenby (1991). Rings, Fields and Groups. Butterworth-Heinemann.
ISBN 0-340-54440-6.
Atiyah M. F., Macdonald, I. G., Introduction to commutative algebra.
Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969
ix+128 pp.
Beachy, J. A. Introductory Lectures on Rings and Modules. Cambridge,
England: Cambridge University Press, 1999.
T.S. Blyth and E.F. Robertson (1985). Groups, rings and fields: Algebra
through practice, Book 3. Cambridge university Press. ISBN 0-521-27288-2.
Dresden, G. "Small Rings." [2]
Ellis, G. Rings and Fields. Oxford, England: Oxford University Press, 1993.
Goodearl, K. R., Warfield, R. B., Jr., An introduction to noncommutative
Noetherian rings. London Mathematical Society Student Texts, 16. Cambridge
University Press, Cambridge, 1989. xviii+303 pp. ISBN 0-521-36086-2
Herstein, I. N., Noncommutative rings. Reprint of the 1968 original. With an
afterword by Lance W. Small. Carus Mathematical Monographs, 15. Mathematical
Association of America, Washington, DC, 1994. xii+202 pp. ISBN 0-88385-015-X
Nagell, T. "Moduls, Rings, and Fields." �6 in Introduction to Number Theory.
New York: Wiley, pp. 19-21, 1951
Nathan Jacobson, Structure of rings. American Mathematical Society
Colloquium Publications, Vol. 37. Revised edition American Mathematical
Society, Providence, R.I. 1964 ix+299 pp.
Nathan Jacobson, The Theory of Rings. American Mathematical Society
Mathematical Surveys, vol. I. American Mathematical Society, New York, 1943.
vi+150 pp.
Kaplansky, Irving (1974), Commutative rings (Revised ed.), University of
Chicago Press, MR0345945
Lam, T. Y., A first course in noncommutative rings. Second edition. Graduate
Texts in Mathematics, 131. Springer-Verlag, New York, 2001. xx+385 pp. ISBN
0-387-95183-0
Lam, T. Y., Exercises in classical ring theory. Second edition. Problem
Books in Mathematics. Springer-Verlag, New York, 2003. xx+359 pp. ISBN
0-387-00500-5
Lam, T. Y., Lectures on modules and rings. Graduate Texts in Mathematics,
189. Springer-Verlag, New York, 1999. xxiv+557 pp. ISBN 0-387-98428-3
Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised
third ed.), New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4
Lang, Serge (2005), Undergraduate Algebra (3rd ed.), Berlin, New York:
Springer-Verlag, ISBN 978-0-387-22025-3 .
Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in
Advanced Mathematics (2nd ed.), Cambridge University Press, ISBN
978-0-521-36764-6
McConnell, J. C.; Robson, J. C. Noncommutative Noetherian rings. Revised
edition. Graduate Studies in Mathematics, 30. American Mathematical Society,
Providence, RI, 2001. xx+636 pp. ISBN 0-8218-2169-5
Pinter-Lucke, James (2007), "Commutativity conditions for rings: 1950-2005",
Expositiones Mathematicae 25 (2): 165-174, doi:10.1016/j.exmath.2006.07.001,
ISSN 0723-0869
Rowen, Louis H., Ring theory. Vol. I, II. Pure and Applied Mathematics, 127,
128. Academic Press, Inc., Boston, MA, 1988. ISBN 0-12-599841-4, ISBN
0-12-599842-2
Sloane, N. J. A. Sequences A027623 and A037234 in "The On-Line Encyclopedia
of Integer Sequences
Zwillinger, D. (Ed.). "Rings." �2.6.3 in CRC Standard Mathematical Tables
and Formulae. Boca Raton, FL: CRC Press, pp. 141-143, 1995
[edit] Special references
Balcerzyk, Stanislaw; J�zefiak, Tadeusz (1989), Commutative Noetherian and
Krull rings, Ellis Horwood Series: Mathematics and its Applications,
Chichester: Ellis Horwood Ltd., ISBN 978-0-13-155615-7
Balcerzyk, Stanislaw; J�zefiak, Tadeusz (1989), Dimension, multiplicity and
homological methods, Ellis Horwood Series: Mathematics and its
Applications., Chichester: Ellis Horwood Ltd., ISBN 978-0-13-155623-2
Ballieu, R. "Anneaux finis; syst�mes hypercomplexes de rang trois sur un
corps commutatif." Ann. Soc. Sci. Bruxelles. S�r. I 61, 222-227, 1947.
Berrick, A. J. and Keating, M. E. An Introduction to Rings and Modules with
K-Theory in View. Cambridge, England: Cambridge University Press, 2000.
Eisenbud, David (1995), Commutative algebra. With a view toward algebraic
geometry., Graduate Texts in Mathematics, 150, Berlin, New York:
Springer-Verlag, MR1322960, ISBN 978-0-387-94268-1; 978-0-387-94269-8
Fine, B. "Classification of Finite Rings of Order ." Math. Mag. 66, 248-252,
1993
Fletcher, C. R. "Rings of Small Order." Math. Gaz. 64, 9-22, 1980
Fraenkel, A. "�ber die Teiler der Null und die Zerlegung von Ringen." J.
reine angew. Math. 145, 139-176, 1914
Gilmer, R. and Mott, J. "Associative Rings of Order ." Proc. Japan Acad. 49,
795-799, 1973
Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational
Science. New York: Springer-Verlag, 1998
Jacobson, Nathan (1945), "Structure theory of algebraic algebras of bounded
degree", Annals of Mathematics 46 (4): 695-707, ISSN 0003-486X
Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical
Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1998
Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and
Engineers. New York: Dover, 2000
Nagata, Masayoshi (1962), Local rings, Interscience Tracts in Pure and
Applied Mathematics, 13, Interscience Publishers, pp. xiii+234, MR0155856,
ISBN 978-0-88275-228-0 (1975 reprint)
Pierce, Richard S., Associative algebras. Graduate Texts in Mathematics, 88.
Studies in the History of Modern Science, 9. Springer-Verlag, New
York-Berlin, 1982. xii+436 pp. ISBN 0-387-90693-2
Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra, Graduate Texts
in Mathematics, 28, 29, Berlin, New York: Springer-Verlag
[edit] Historical references
History of ring theory at the MacTutor Archive
Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York:
Macmillian, 1996
Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 4th ed.
New York: Springer-Verlag, 2004. ISBN 3-540-43491-7
Faith, Carl, Rings and things and a fine array of twentieth century
associative algebra. Mathematical Surveys and Monographs, 65. American
Mathematical Society, Providence, RI, 1999. xxxiv+422 pp. ISBN 0-8218-0993-8
It�, K. (Ed.). "Rings." �368 in Encyclopedic Dictionary of Mathematics, 2nd
ed., Vol. 2. Cambridge, MA: MIT Press, 1986
Kleiner, I. "The Genesis of the Abstract Ring Concept." Amer. Math. Monthly
103, 417-424, 1996
Renteln, P. and Dundes, A. "Foolproof: A Sampling of Mathematical Folk
Humor." Notices Amer. Math. Soc. 52, 24-34, 2005
Singmaster, D. and Bloom, D. M. "Problem E1648." Amer. Math. Monthly 71,
918-920, 1964
Van der Waerden, B. L. A History of Algebra. New York: Springer-Verlag, 1985
Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168,
2002
Retrieved from "http://en.wikipedia.org/wiki/Ring_(mathematics)"
Categories: Algebraic structures | Mathematical structures | Ring theory


From: Generalzod on 9 Jan 2010 14:55

"JSH" <jstevh(a)gmail.com> wrote in message
news:a19a155b-e5e0-4321-895a-96f7a4f944cb(a)a21g2000yqc.googlegroups.com...
> One of the hardest things for me to understand has not been the math.
> The mathematics of the flaw in modern number theory is kind of
> convoluted but can be worked out and relies as its linchpin on the
> distributive property, so it is resolvable. The weirder thing for me
> has been understanding first how mathematicians could go on with the
> error, and now how physicists might as well, even though as a result
> they know their theory is wrong, and that theory doesn't work! So
> maybe some of them are sabotaging physics experiments like the LHC.

LCH is evil incarnate;
http://www.youtube.com/watch?v=Lt1Yo610lG0>



> James Harris
>
>
>
>
>


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