# Glossary of ring theory

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{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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.

## 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,*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*.

## Types of elements

**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*r*^{2}=*r*.

**integral element**- For 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*r*^{n}= 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*^{−1}*r*= 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*a*_{1}, ...,*a*_{n}such that*I*=*Ra*_{1}+ ... +*Ra*_{n}. A right ideal*I*is*finitely generated*if there exist finitely many elements*a*_{1}, ...,*a*_{n}such that*I*=*a*_{1}*R*+ ... +*a*_{n}*R*. A two-sided ideal*I*is*finitely generated*if there exist finitely many elements*a*_{1}, ...,*a*_{n}such that*I*=*Ra*_{1}*R*+ ... +*Ra*_{n}*R*.

**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*I*^{k}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) = 1

- for 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 ideal**- Every 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*.

## Types of rings

**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 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 ring**- a ring that is finitely generated as
**Z**-algebra.

**Finitely presented algebra**- If
*R*is a commutative ring and*A*is an*R*-algebra, then*A*is a**finitely presented**if it is a quotient of a polynomial ring over*R*-algebra*R*in finitely many variables by a finitely generated ideal.^{[1]}

**hereditary ring**- A 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 number**- A ring
*R*has*invariant basis number*if*R*^{m}isomorphic to*R*^{n}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 ring**- See rng of square zero.

**perfect ring**- A
*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 zero**- A 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 ring**- A ring
*R*is*left self-injective*if the module_{R}*R*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 ring**- This 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 ring**- The ring consisting of a single element 0 = 1, also called the zero ring.

**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*. This essentially means that every non-zero non-unit can be written uniquely as a product of irreducible elements.

**von Neumann regular ring**- A 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 ring**- The 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.

## Ring constructions

**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 ring**- Given 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 ring
- Given a ring
*R*, its opposite ring*R*^{op}has the same underlying set as*R*, the addition operation is defined as in*R*, but the product of*s*and*r*in*R*^{op}is*rs*, while the product is*sr*in*R*.

- projective line over a ring
- Given a ring
*R*, its projective line P(*R*) provides the context for linear fractional transformations of*R*.

### Polynomial rings

{{#invoke:main|main}}

- formal power series ring

- Laurent polynomial ring

- polynomial ring
- Given
*R*a commutative ring. The polynomial ring*R*[*x*] is defined to be the set with addition defined by , and with multiplication defined by . - 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*].

- If

- ring of rational functions

- skew polynomial ring
- Given
*R*a ring and an endomorphism of*R*. The skew polynomial ring is defined to be the set , with addition defined as usual, and multiplication defined by the relation .

## Miscellaneous

**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.

## Ringlike structures

The following structures include generalizations and other algebraic objects similar to rings.

**nearring**- A 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
**i**ng" without an "**i**dentity".

**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.