Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... In mathematics, multiplication is an elementary arithmetic operation. ...

Definition of a ring

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, a monoid under multiplication, and such that multiplication is both left and right distributive over addition. Note that rings are assumed to have multiplicative identities unless otherwise noted. The additive identity is denoted by 0 and the multiplicative identity by 1.

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.

In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ...

Types of elements

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.

Idempotent 

An element e of a ring is idempotent if e² = e.

Irreducible 

An element x of a ring is irreducible if it is not a unit and for any elements a and b such that x=a b, either a or b is a unit. Note that every prime is irreducible, but not necessarily vice versa.

Prime 

An element x of a ring is prime if it is not a unit and for any elements a, b ≠ 1 such that x=a b, either 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.

Zero divisor 

A nonzero element r of R is said to be a zero divisor if there exists s ≠ 0 such that sr=0 or rs=0. If a ring has a Zero divisor which is also a unit, then the ring has no other elements and is the trivial ring.

The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ... In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... This picture illustrates how the hours in a clock form a group. ... In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ...

Homomorphisms and ideals

Factor ring 

Given a ring R and an ideal I of R, the factor ring is the set R/I of cosets {a+I : a∈R} together with 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.

Finitely generated ideal 

A left ideal I is finitely generated if there exist finitely many elements a₁,...,a_n such that I = Ra₁ + ... + Ra_n. A right ideal I is finitely generated if there exist finitely many elements a₁,...,a_n such that I = a₁R + ... + a_nR. A two-sided ideal I is finitely generated if there exist finitely many elements a₁,...,a_n such that I = Ra₁R + ... + Ra_nR.

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 for emphasis: a two-sided ideal) 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 is the preimage of the element 0 of the codomain of the ring homomorphism. 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. A nil ring would be a ring consisting only of nilpotent elements; under the convention that rings are unital the only example would be the zero ring, but any rng (q.v.) that would be nil in that sense becomes a nil ideal inside the ring formed by adding 1. Nil algebras similarly.

Nilpotent ideal 

An ideal I is nilpotent if its powers I^k are {0} for k large enough. Nilpotent implies nil, but not conversely. See the remarks at nil ideal for nilpotent rings or algebras. Locally nilpotent for a ring means finitely-generated subrings (subrings, really, since 1 is not required) are nilpotent.

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 Prime Ideals. It is 'not' equal, in general, to the 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 the 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.

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 has the special properties that

f(a + b) = f(a) ⊕ f(b)

f(a * b) = f(a) × f(b)

f(1) = 1

for any elements a and b of R.

Ring monomorphism 

A ring homomorphism that is injective is a ring monomorphism.

Ring epimorphism 

A ring homomorphism that is surjective is a ring epimorphism.

Ring isomorphism 

A ring homomorphism that is bijective is a ring isomorphism. The inverse of an isomorphism, it turns out, 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.

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In abstract algebra, for a number of algebraic structures, the fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements of R which in a sense are close to zero. // The Jacobson radical is denoted by J(R) and can be defined in the following equivalent... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ... In mathematics, more specifically in ring theory a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals, i. ... In ring theory, a branch of mathematics, the radical of a ring isolates certain bad properties of the ring. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In Ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R. More specifically: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R... In ring theory, a branch of mathematics, the radical of an ideal is a kind of completion of the ideal. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... Partial plot of a function f. ... In abstract algebra, a ring monomorphism is an injective ring homomorphism (a function between two rings which respects the operations of addition and multiplication). ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In abstract algebra, a ring epimorphism is a surjective ring homomorphism (a function between two rings which respects the operations of addition and multiplication). ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In abstract algebra, a ring isomorphism is a bijective ring homomorphism (a function between two rings which respects the operations of addition and multiplication). ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

Types of rings

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 idempotent 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 

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 (ring theory) 

A ring without zero divisors and in which 1≠0. The noncommutative generalization of integral domain.

Euclidean domain 

An integral domain in which a degree function is defined so that "division with remainder" can be carried out is called a Euclidean domain (because the Euclidean algorithm works in these rings). All Euclidean domains are principal ideal domains.

Field 

A commutative division ring is a field. Every finite division ring is a field, as is every finite integral domain. Field theory is in fact an older branch of mathematics than ring theory.

Integral domain or entire ring 

A commutative ring without zero divisors and in which 1≠0 is an integral domain.

Invariant basis number

A ring R has invariant basis number if R^m 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. Any ring can be made local via localization.

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.

Prime ring 

A common generalization of both simple rings and domains.

Primitive ring 

A generalization of simple rings. Primitive rings are prime.

Semisimple ring 

A ring that has a "nice" decomposition. A semisimple ring is also Noetherian, and has no nilpotent ideals. A ring can be made semi-simple if it is divided by its Jacobson radical.

Simple ring 

A ring with no two-sided ideals.

Unique factorization domain or factorial ring

An integral domain R in which every non-zero non-unit element can be written as a product of prime elements of R.

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.

In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. ... In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a1 ≤ a2 ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that am = an for all m > n. ... In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists of idempotent elements. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a Dedekind domain is a Noetherian integral domain which is integrally closed in its fraction field and which has Krull dimension 1. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In abstract algebra, a domain is the noncommutative analogue of an integral domain. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ... In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Field theory is a branch of mathematics which studies the properties of fields. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ... in mathematics, the invariant basis number (IBN) property of a ring R is the property that all free modules over R are similarly well-behaved to vector spaces with respect to the uniqueness of their ranks. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ... Localization can mean any of the following: Generally, localization is the determination of the locality (position) of an object. ... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ... In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a1 ≤ a2 ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that am = an for all m > n. ... In abstract algebra, a ring R is a prime ring if for any two elements a and b of R, if arb = 0 for all r in R, then either a = 0 or b = 0. ... In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In abstract algebra, a left primitive ring R is a ring with a faithful simple left module R-module. ... In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. ... In abstract algebra, a ring R is a prime ring if for any two elements a and b of R, if arb = 0 for all r in R, then either a = 0 or b = 0. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... In ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements of R which in a sense are close to zero. // The Jacobson radical is denoted by J(R) and can be defined in the following equivalent... In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. ... In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ...

Miscellaneous

Characteristic 

The characteristic of a ring is the smallest positive integer n satisfying n1 = 0 if it exists and 0 otherwise. As a consequence, nx=0 for all elements x of the ring.

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 operation component-wise.

Krull dimension of a commutative ring 

The maximal length of a strictly increasing chain of prime ideals in the ring.

Localization of a ring 

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.

Rng 

A rng is 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 "ring" without an "identity".

Semiring 

A semiring is 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. That is, elements in a semiring need not have additive inverses.