ring A
ring is a
set R with two
binary operations, usually called addition (+) and multiplication (×), such that
R is an
abelian group under addition,
R is a
monoid under multiplication, and multiplication is both left and right
distributive over addition. Rings are assumed to have multiplicative identities unless otherwise noted. The additive identity is denoted by 0 and the multiplicative identity by 1. (
Warning: some books, especially older books, use the term "ring" to mean what here will be called a
rng; i.e., they do not require a ring to have a multiplicative identity.)
subring A subset
S of the ring (
R,+,×) which remains a ring when + and × are restricted to
S and contains the multiplicative identity 1 of
R is called a
subring of
R.
associate In a
commutative ring, an element
a is called an
associate of an element
b if
a divides
b and
b divides
a.
central An element
r of a ring
R is
central if
xr = rx for all
x in
R. The
set of all central elements forms a
subring of
R, known as the
center of
R.
divisor In an
integral domain R, an element
a is called a
divisor of the element
b (and we say
a divides b) if there exists an element
x in
R with
ax = b.
idempotent An element
r of a ring is
idempotent if
r2 = r.
integral elementFor a
commutative ring B containing a subring
A, an element
b is
integral over A if it satisfies a monic polynomial with coefficients from
A.
irreducible An element
x of an integral
domain is
irreducible if it is not a
unit and for any elements
a and
b such that
x = ab, either
a or
b is a
unit. Note that every
prime element is irreducible, but not necessarily vice versa.
prime element An element
x of an
integral domain is a
prime element if it is not zero and not a unit and whenever
x divides a product
ab,
x divides
a or
x divides
b.
nilpotent An element
r of
R is
nilpotent if there exists a positive integer
n such that
rn = 0.
unit or invertible element An element
r of the ring
R is a
unit if there exists an element
r−1 such that
rr−1 = r−1r = 1. This element
r−1 is uniquely determined by
r and is called the
multiplicative inverse of
r. The set of units forms a
group under multiplication.
von Neumann regular element An element
r of a ring
R is
von Neumann regular if there exists an element
x of
R such that
r = rxr.
zero divisor An element
r of
R is a
left zero divisor if there exists a nonzero element
x in
R such that
rx = 0 and a
right zero divisor or if there exists a nonzero element
y in
R such that
yr = 0. An element
r of
R is a called a
two-sided zero divisor if it is both a left zero divisor and a right zero divisor.
Homomorphisms and ideals
finitely generated ideal A left ideal
I is
finitely generated if there exist finitely many elements
a1, ..., an such that
I = Ra1 + ... + Ran. A right ideal
I is
finitely generated if there exist finitely many elements
a1, ..., an such that
I = a1R + ... + anR. A two-sided ideal
I is
finitely generated if there exist finitely many elements
a1, ..., an such that
I = Ra1R + ... + RanR.
ideal A
left ideal I of
R is a subgroup of
R such that
aI ⊆ I for all
a ∈ R. A
right ideal is a subgroup of
R such that
Ia ⊆ I for all
a ∈ R. An
ideal (sometimes called a
two-sided ideal for emphasis) is a subgroup which is both a left ideal and a right ideal.
Jacobson radical The intersection of all maximal left ideals in a ring forms a two-sided ideal, the
Jacobson radical of the ring.
kernel of a ring homomorphism The
kernel of a
ring homomorphism f : R → S is the set of all elements
x of
R such that
f(x) = 0. Every ideal is the kernel of a ring homomorphism and vice versa.
maximal ideal A left ideal
M of the ring
R is a
maximal left ideal if
M ≠ R and the only left ideals containing
M are
R and
M itself. Maximal right ideals are defined similarly. In commutative rings, there is no difference, and one speaks simply of
maximal ideals.
nil ideal An ideal is
nil if it consists only of nilpotent elements.
nilpotent ideal An ideal
I is
nilpotent if the power
Ik is {0} for some positive integer
k. Every
nilpotent ideal is nil, but the converse is not true in general.
nilradical The set of all nilpotent elements in a commutative ring forms an ideal, the
nilradical of the ring. The nilradical is equal to the intersection of all the ring's
prime ideals. It is contained in, but in general not equal to, the ring's
Jacobson radical.
prime ideal An ideal
P in a commutative ring
R is
prime if
P ≠ R and if for all
a and
b in
R with
ab in
P, we have
a in
P or
b in
P. Every maximal ideal in a commutative ring is prime. There is also a definition of
prime ideal for noncommutative rings.
principal ideal A
principal left ideal in a ring
R is a left ideal of the form
Ra for some element
a of
R. A
principal right ideal is a right ideal of the form
aR for some element
a of
R. A
principal ideal is a two-sided ideal of the form
RaR for some element
a of
R.
quotient ring or factor ring Given a ring
R and an ideal
I of
R, the
quotient ring is the ring formed by the set
R/
I of
cosets {a + I : a∈R} together with the operations
(a + I) + (b + I) = (a + b) + I and
(a + I)(b + I) = ab + I. The relationship between ideals, homomorphisms, and factor rings is summed up in the
fundamental theorem on homomorphisms.
radical of an ideal The
radical of an ideal I in a commutative ring consists of all those ring elements a power of which lies in
I. It is equal to the intersection of all prime ideals containing
I.
ring homomorphism A
function f : R → S between rings
(R, +, ∗) and
(S, ⊕, ×) is a
ring homomorphism if it satisfies
f(
a +
b) =
f(
a) ⊕
f(
b)
f(
a ∗
b) =
f(
a) ×
f(
b)
f(1) = 1for all elements
a and
b of
R.
ring monomorphism A ring homomorphism that is injective is a
ring monomorphism.
ring isomorphism A ring homomorphism that is bijective is a
ring isomorphism. The inverse of a ring isomorphism is also a ring isomorphism. Two rings are
isomorphic if there exists a ring isomorphism between them. Isomorphic rings can be thought as essentially the same, only with different labels on the individual elements.
trivial idealEvery
nonzero ring R is guaranteed to have two ideals: the zero ideal and the entire ring
R. These ideals are usually referred to as
trivial ideals. Right ideals, left ideals, and two-sided ideals other than these are called
nontrivial.
Abelian ring A ring in which all idempotent elements are central is called an Abelian ring. Such rings need not be commutative.
artinian ring A ring satisfying the descending chain condition for left ideals is
left artinian; if it satisfies the descending chain condition for right ideals, it is
right artinian; if it is both left and right artinian, it is called
artinian. Artinian rings are noetherian.
boolean ring A ring in which every element is multiplicatively idempotent element is a
boolean ring.
commutative ring A ring
R is
commutative if the multiplication is commutative, i.e.
rs = sr for all
r,s ∈ R.
Dedekind domain A
Dedekind domain is an integral domain in which every ideal has a unique factorization into prime ideals.
division ring or skew field A ring in which every nonzero element is a unit and
1 ≠ 0 is a
division ring.
domain A
domain is a nonzero ring with no
zero divisors except 0. This is the noncommutative generalization of integral domain.
Euclidean domain A
Euclidean domain is an integral domain in which a degree function is defined so that "division with remainder" can be carried out. It is so named because the Euclidean algorithm is a well-defined algorithm in these rings. All Euclidean domains are principal ideal domains.
field A
field is a commutative division ring. Every finite division ring is a field, as is every finite integral domain.
finitely generated ringa ring that is finitely generated as
Z-algebra.
Finitely presented algebraIf
R is a commutative ring and
A is an
R-algebra, then
A is a
finitely presented R-algebra if it is a quotient of a
polynomial ring over
R in finitely many variables by a finitely generated ideal.
hereditary ringA ring is
left hereditary if its left ideals are all projective modules. Right hereditary rings are defined analogously.
integral domain or entire ring A nonzero commutative ring with no zero divisors except 0.
invariant basis numberA ring
R has
invariant basis number if
Rm isomorphic to
Rn as
R-modules implies
m = n.
local ring A ring with a unique maximal left ideal is a
local ring. These rings also have a unique maximal right ideal, and the left and the right unique maximal ideals coincide. Certain commutative rings can be embedded in local rings via localization at a prime ideal.
Noetherian ring A ring satisfying the
ascending chain condition for left ideals is
left Noetherian; a ring satisfying the ascending chain condition for right ideals is
right Noetherian; a ring that is both left and right Noetherian is
Noetherian. A ring is left Noetherian if and only if all its left ideals are finitely generated; analogously for right Noetherian rings.
null ringSee
rng of square zero.
perfect ringA
left perfect ring is one satisfying the descending chain condition on
right principal ideals. They are also characterized as rings whose flat left modules are all projective modules. Right perfect rings are defined analogously. Artinian rings are perfect.
prime ring A nonzero ring
R is called a
prime ring if for any two elements
a and
b of
R with
aRb = 0, we have either
a = 0 or
b = 0. This is equivalent to saying that the zero ideal is a prime ideal. Every
simple ring and every domain is a prime ring.
primitive ring A
left primitive ring is a ring that has a faithful
simple left
R-module. Every
simple ring is
primitive. Primitive rings are prime.
principal ideal domain An integral domain in which every ideal is principal is a
principal ideal domain. All principal ideal domains are unique factorization domains.
quasi-Frobenius ring a special type of
Artinian ring which is also a self-injective ring on both sides. Every
semisimple ring is quasi-Frobenius.
rng of square zeroA rng in which
xy = 0 for all
x and
y. These are sometimes also called
zero rings, even though they usually do not have a 1.
self-injective ringA ring
R is
left self-injective if the module
RR is an
injective module. While rings with unity are always projective as modules, they are not always injective as modules.
semiprimitive ring or Jacobson semisimple ringThis is a ring whose Jacobson radical is zero. Von Neumann regular rings and primitive rings are semiprimitive, however quasi-Frobenius rings and local rings are usually not semiprimitive.
semisimple ring A
semisimple ring is a ring
R that has a "nice" decomposition, in the sense that
R is a semisimple left
R-module. Every semisimple ring is also Artinian, and has no nilpotent ideals. The Artin–Wedderburn theorem asserts that every semisimple ring is a finite product of full matrix rings over division rings.
simple ring A non-
zero ring which only has trivial two-sided ideals (the zero ideal, the ring itself, and no more) is a
simple ring.
trivial ringThe ring consisting of a single element
0 = 1, also called the zero ring.
unique factorization domain or factorial ringAn integral domain
R in which every non-zero non-unit element can be written as a product of
prime elements of
R. This essentially means that every non-zero non-unit can be written uniquely as a product of irreducible elements.
von Neumann regular ringA ring for which each element
a can be expressed as
a = axa for another element
x in the ring. Semisimple rings are von Neumann regular.
zero ringThe ring consisting only of a single element
0 = 1, also called the trivial ring. Sometimes "zero ring" is alternatively used to mean rng of square zero.
direct product of a family of rings This is a way to construct a new ring from given rings by taking the
cartesian product of the given rings and defining the algebraic operations component-wise.
endomorphism ring A ring formed by the
endomorphisms of an algebraic structure. Usually its multiplication is taken to be
function composition, while its addition is pointwise addition of the images.
localization of a ring For commutative rings, a technique to turn a given set of elements of a ring into units. It is named
Localization because it can be used to make any given ring into a
local ring. To localize a ring
R, take a multiplicatively closed subset
S containing no zero divisors, and formally define their multiplicative inverses, which shall be added into
R. Localization in noncommutative rings is more complicated, and has been in defined several different ways.
matrix ringGiven a ring
R, it is possible to construct
matrix rings whose entries come from
R. Often these are the square matrix rings, but under certain conditions "infinite matrix rings" are also possible. Square matrix rings arise as endomorphism rings of free modules with finite rank.
opposite ringGiven a ring
R, its opposite ring
Rop has the same underlying set as
R, the addition operation is defined as in
R, but the product of
s and
r in
Rop is
rs, while the product is
sr in
R.
projective line over a ringGiven a ring
R, its projective line P(
R) provides the context for linear fractional transformations of
R.
differential polynomial ringformal power series ringLaurent polynomial ringmonoid ringpolynomial ringGiven
R a commutative ring. The polynomial ring
R[
x] is defined to be the set
{ a n x n + a n − 1 x n − 1 + a n − 2 x n − 2 + … + a 1 x + a 0 | a n , a n − 1 , a n − 2 , … , a 0 ∈ R } with addition defined by
( ∑ i a i x i ) + ( ∑ i b i x i ) = ∑ i ( a i + b i ) x i , and with multiplication defined by
( ∑ i a i x i ) ⋅ ( ∑ j b j x j ) = ∑ k ( ∑ i , j : i + j = k a i b j ) x k .Some results about properties of
R and
R[
x]:
If R is UFD, so is R[x].If R is Noetherian, so is R[x].ring of rational functionsskew polynomial ringGiven
R a ring and an endomorphism
σ ∈ End ( R ) of
R. The skew polynomial ring
R [ x ; σ ] is defined to be the set
{ a n x n + a n − 1 x n − 1 + a n − 2 x n − 2 + … + a 1 x + a 0 | a n , a n − 1 , a n − 2 , … , a 0 ∈ R } , with addition defined as usual, and multiplication defined by the relation
x a = σ ( a ) x ∀ a ∈ R .
characteristic The
characteristic of a ring is the smallest positive integer
n satisfying
nx = 0 for all elements
x of the ring, if such an
n exists. Otherwise, the characteristic is 0.
Krull dimension of a commutative ring The maximal length of a strictly increasing chain of prime ideals in the ring.
The following structures include generalizations and other algebraic objects similar to rings.
nearringA structure that is a group under addition, a
semigroup under multiplication, and whose multiplication distributes on the right over addition.
rng (or pseudo-ring)An algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element. The term "rng" is meant to suggest that it is a "r
ing" without an "
identity".
semiring An algebraic structure satisfying the same properties as a ring, except that addition need only be an abelian monoid operation, rather than an
abelian group operation. That is, elements in a
semiring need not have additive inverses.
