For specific types of algebras, see also:Glossary of field theory andGlossary of Lie groups and Lie algebras. Since, currently, there is no glossary on not-necessarily-associative algebra structures in general, this glossary includes some concepts that do not need associativity; e.g., a derivation.
TheAmitsur complex of a ring homomorphism is a cochain complex that measures the extent in which the ring homomorphism fails to befaithfully flat.
Artinian
A leftArtinian ring is a ring satisfying thedescending chain condition for left ideals; a right Artinian ring is one satisfying the descending chain condition for right ideals. If a ring is both left and right Artinian, it is calledArtinian. Artinian rings are Noetherian rings.
associate
In a commutative ring, an elementa is called anassociate of an elementb ifa dividesb andb dividesa.
automorphism
Aring automorphism is a ring isomorphism between the same ring; in other words, it is a unit element of the endomorphism ring of the ring that is multiplicative and preserves the multiplicative identity.
Analgebra automorphism over a commutative ringR is an algebra isomorphism between the same algebra; it is a ring automorphism that is alsoR-linear.
Azumaya
AnAzumaya algebra is a generalization of a central simple algebra to a non-field base ring.
Thebidimension of an associative algebraA over a commutative ringR is the projective dimension ofA as an(Aop ⊗RA)-module. For example, an algebra has bidimension zero if and only if it is separable.
1. Thecentralizer of a subsetS of a ring is the subring of the ring consisting of the elements commuting with the elements ofS. For example, the centralizer of the ring itself is the centre of the ring.
1. Thecharacteristic of a ring is the smallest positive integern satisfyingnx = 0 for all elementsx of the ring, if such ann exists. Otherwise, the characteristic is 0.
2. Thecharacteristic subring ofR is the smallest subring (i.e., the unique minimal subring). It is necessary the image of the unique ring homomorphismZ →R and thus is isomorphic toZ/n wheren is the characteristic ofR.
change
Achange of rings is a functor (between appropriate categories) induced by a ring homomorphism.
Clifford algebra
AClifford algebra is a certain associative algebra that is useful in geometry and physics.
coherent
A leftcoherent ring is a ring such that every finitely generated left ideal of it is a finitely presented module; in other words, it iscoherent as a left module over itself.
commutative
1. A ringR iscommutative if the multiplication is commutative, i.e.rs =sr for allr,s ∈R.
2. A ringR isskew-commutative ring ifxy = (−1)ε(x)ε(y)yx, whereε(x) denotes the parity of an elementx.
3. A commutative algebra is an associative algebra that is a commutative ring.
Adirect product of a family of rings is a ring given by taking thecartesian product of the given rings and defining the algebraic operations component-wise.
2. An elementr ofR is aleftzero divisor if there exists a nonzero elementx inR such thatrx = 0 and aright zero divisor or if there exists a nonzero elementy inR such thatyr = 0. An elementr ofR is a called atwo-sided zero divisor if it is both a left zero divisor and a right zero divisor.
division
Adivision ring or skew field is a ring in which every nonzero element is a unit and1 ≠ 0.
domain
Adomain is a nonzero ring with no zero divisors except 0. For a historical reason, a commutative domain is called anintegral domain.
Anendomorphism ring is a ring formed by theendomorphisms of an object with additive structure; the multiplication is taken to befunction composition, while its addition is pointwise addition of the images.
enveloping algebra
The (universal)enveloping algebraE of a not-necessarily-associative algebraA is the associative algebra determined byA in some universal way. The best known example is theuniversal enveloping algebra of a Lie algebra.
1. A left idealI isfinitely generated if there exist finitely many elementsa1, ...,an such thatI =Ra1 + ... +Ran. A right idealI isfinitely generated if there exist finitely many elementsa1, ...,an such thatI =a1R + ... +anR. A two-sided idealI isfinitely generated if there exist finitely many elementsa1, ...,an such thatI =Ra1R + ... +RanR.
1. Afree ideal ring or a fir is a ring in which every right ideal is a free module of fixed rank.
2. A semifir is a ring in which every finitely generated right ideal is a free module of fixed rank.
3. Thefree product of a family of associative is an associative algebra obtained, roughly, by the generators and the relations of the algebras in the family. The notion depends on which category of associative algebra is considered; for example, in the category of commutative rings, a free product is a tensor product.
Agraded ring is a ring together with a grading or a graduation; i.e, it is a direct sum of additive subgroups with the multiplication that respects the grading. For example, a polynomial ring is a graded ring by degrees of polynomials.
generate
An associative algebraA over a commutative ringR is said to begenerated by a subsetS ofA if the smallest subalgebra containingS isA itself andS is said to be the generating set ofA. If there is a finite generating set,A is said to be afinitely generated algebra.
Aleft idealI ofR is an additive subgroup ofR such thataI ⊆I for alla ∈R. Aright ideal is a subgroup ofR such thatIa ⊆I for alla ∈R. Anideal (sometimes called atwo-sided ideal for emphasis) is a subgroup that is both a left ideal and a right ideal.
An elementx of an integral domain isirreducible if it is not a unit and for any elementsa andb such thatx =ab, eithera orb is a unit. Note that every prime element is irreducible, but not necessarily vice versa.
Thekernel of a ring homomorphism of a ring homomorphismf :R →S is the set of all elementsx ofR such thatf(x) = 0. Every ideal is the kernel of a ring homomorphism and vice versa.
Köthe
Köthe's conjecture states that if a ring has a nonzero nil right ideal, then it has a nonzero nil ideal.
1. A ring with a unique maximal left ideal is alocal 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 vialocalization at aprime ideal.
2. Alocalization of a ring : For commutative rings, a technique to turn a given set of elements of a ring into units. It is namedLocalization because it can be used to make any given ring into alocal ring. To localize a ringR, take a multiplicatively closed subsetS that contains nozero divisors, and formally define their multiplicative inverses, which are then added intoR. Localization in noncommutative rings is more complicated, and has been in defined several different ways.
1. A left idealM of the ringR is amaximal left ideal (resp. minimal left ideal) if it is maximal (resp. minimal) among proper (resp. nonzero) left ideals. Maximal (resp. minimal) right ideals are defined similarly.
2. Amaximal subring is a subring that is maximal among proper subrings. A "minimal subring" can be defined analogously; it is unique and is called thecharacteristic subring.
matrix
1. Amatrix ring over a ringR is a ring whose elements are square matrices of fixed size with the entries inR. The matrix ring or the full matrix ring of matrices overR isthe matrix ring consisting of all square matrices of fixed size with the entries inR. When the grammatical construction is not workable, the term "matrix ring" often refers to the "full" matrix ring when the context makes no confusion likely; for example, when one says a semsimple ring is a product of matrix rings of division rings, it is implicitly assumed that "matrix rings" refer to "full matrix rings". Every ring is (isomorphic to) the full matrix ring over itself.
2. Thering of generic matrices is the ring consisting of square matrices with entries in formal variables.
Anearring is a structure that is a group under addition, asemigroup under multiplication, and whose multiplication distributes on the right over addition.
nil
1. Anil ideal is an ideal consisting of nilpotent elements.
3. The (Baer)lower nil radical is the intersection of all prime ideals. For a commutative ring, the upper nil radical and the lower nil radical coincide.
nilpotent
1. An elementr ofR isnilpotent if there exists a positive integern such thatrn = 0.
2. Anil ideal is an ideal whose elements are nilpotent elements.
3. Anilpotent ideal is an ideal whosepowerIk is {0} for some positive integerk. Every nilpotent ideal is nil, but the converse is not true in general.
4. Thenilradical of a commutative ring is the ideal that consists of all nilpotent elements of the ring. It is equal to the intersection of all the ring'sprime ideals and is contained in, but in general not equal to, the ring's Jacobson radical.
Noetherian
A leftNoetherian ring is a ring satisfying theascending chain condition for left ideals. Aright Noetherian is defined similarly and a ring that is both left and right Noetherian isNoetherian. A ring is left Noetherian if and only if all its left ideals are finitely generated; analogously for right Noetherian rings.
Given a ringR, itsopposite ringRop has the same underlying set asR, the addition operation is defined as inR, but the product ofs andr inRop isrs, while the product issr inR.
order
Anorder of an algebra is (roughly) a subalgebra that is also a full lattice.
Ore
A leftOre domain is a (non-commutative) domain for which the set of non-zero elements satisfies the left Ore condition. A right Ore domain is defined similarly.
Aleftperfect ring is one satisfying thedescending chain condition onright 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.
polynomial
1. Apolynomial ring over a commutative ringR is a commutative ring consisting of all the polynomials in the specified variables with coefficients in R.
Given a ringR and an endomorphismσ ∈ End(R) ofR. The skew polynomial ringR[x;σ] is defined to be the set{anxn +an−1xn−1 + ... +a1x +a0 |n ∈N,an,an−1, ...,a1,a0 ∈R}, with addition defined as usual, and multiplication defined by the relationxa =σ(a)x ∀a ∈R.
prime
1. An elementx of an integral domain is aprime element if it is not zero and not a unit and wheneverx divides a productab,x dividesa orx dividesb.
2. An idealP in acommutative ringR isprime ifP ≠R and if for alla andb inR withab inP, we havea inP orb inP. Every maximal ideal in a commutative ring is prime.
3. An idealP in a (not necessarily commutative) ringR is prime ifP ≠R and for all idealsA andB ofR,AB ⊆P impliesA ⊆P orB ⊆P. This extends the definition for commutative rings.
4. prime ring : Anonzero ringR is called aprime ring if for any two elementsa andb ofR withaRb = 0, we have eithera = 0 orb = 0. This is equivalent to saying that the zero ideal is a prime ideal (in the noncommutative sense.) Everysimple ring and everydomain is a prime ring.
2. An idealI of a ringR is said to beprimitive ifR/I is primitive.
principal
Aprincipal ideal : Aprincipal left ideal in a ringR is a left ideal of the formRa for some elementa ofR. Aprincipal right ideal is a right ideal of the formaR for some elementa ofR. Aprincipal ideal is a two-sided ideal of the formRaR for some elementa ofR.
quotient ring orfactor ring : Given a ringR and an idealI ofR, thequotient ring is the ring formed by the setR/I ofcosets{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 thefundamental theorem on homomorphisms.
Theradical of an idealI in acommutative ring consists of all those ring elements a power of which lies inI. It is equal to the intersection of all prime ideals containingI.
ring
1. AsetR with twobinary operations, usually called addition (+) and multiplication (×), such thatR is anabelian group under addition,R is amonoid under multiplication, and multiplication is both left and rightdistributive 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 arng; i.e., they do not require a ring to have a multiplicative identity.)
2. Aring homomorphism : Afunctionf :R →S between rings(R, +, ∗) and(S, ⊕, ×) is aring homomorphism if it satisfies
f(a +b) =f(a) ⊕f(b)
f(a ∗b) =f(a) ×f(b)
f(1) = 1
for all elementsa andb ofR.
3. ring isomorphism : A ring homomorphism that isbijective is aring isomorphism. The inverse of a ring isomorphism is also a ring isomorphism. Two rings areisomorphic 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.
rng
1. Arng is asetR with twobinary operations, usually called addition (+) and multiplication (×), such that(R, +) is anabelian group,(R, ×) is asemigroup, and multiplication is both left and rightdistributive over addition. A rng that has anidentity element is a "ring".
A ringR isleftself-injective if the moduleRR is aninjective module. While rings with unity are always projective as modules, they are not always injective as modules.
semiperfect
Asemiperfect ring is a ringR such that, for the Jacobson radical J(R) ofR, (1)R/J(R) is semisimple and (2) idempotents lift modulo J(R).
semiprimary
Asemiprimary ring is a ringR such that, for the Jacobson radical J(R) ofR, (1)R/J(R) is semisimple and (2) J(R) is anilpotent ideal.
semiprime
1. Asemiprime ring is a ring where the onlynilpotent ideal is the trivial ideal {0}. A commutative ring is semiprime if and only if it is reduced.
2. An idealI of a ringR issemiprime if for any idealA ofR,An ⊆I impliesA ⊆I. Equivalently,I is semiprime if and only ifR/I is a semiprime ring.
semiprimitive
Asemiprimitive ring or Jacobson semisimple ring is a ring whoseJacobson radical is zero. Von Neumann regular rings and primitive rings are semiprimitive, however quasi-Frobenius rings and local rings are usually not semiprimitive.
semiring
Asemiring : An algebraic structure satisfying the same properties as a ring, except that addition need only be an abelianmonoid operation, rather than an abelian group operation. That is, elements in a semiring need not have additive inverses.
semisimple
Asemisimple ring is an Artinian ringR that is a finite product of simple Artinian rings; in other words, it is asemisimple leftR-module.
A rightserial ring is a ring that is a right serial module over itself.
Severi–Brauer
TheSeveri–Brauer variety is an algebraic variety associated to a given central simple algebra.
simple
1. Asimple ring is a non-zero ring that only has trivial two-sided ideals (the zero ideal, the ring itself, and no more) is asimple ring.
2. Asimple algebra is an associative algebra that is a simple ring.
singular submodule
The right (resp. left)R-moduleM has asingular submodule if it consists of elements whoseannihilators areessential right (resp. left)ideals inR. In set notation it is usually denoted asZ(M) = {m ∈M | ann(m) ⊆eR}.
subring
Asubring is a subsetS of the ring(R, +, ×) that remains a ring when + and × are restricted toS and contains the multiplicative identity 1 ofR.
symmetric algebra
1. Thesymmetric algebra of a vector space or a moduleV is the quotient of the tensor algebra ofV by the ideal generated by elements of the formx ⊗y −y ⊗x.
2. Thegraded-symmetric algebra of a vector space or a moduleV is a variant of the symmetric algebra that is constructed by taking grading into account.
unit orinvertible element : An elementr of the ringR is aunit if there exists an elementr−1 such thatrr−1 =r−1r = 1. This elementr−1 is uniquely determined byr and is called themultiplicative inverse ofr. The set of units forms agroup under multiplication.
unity
The term "unity" is another name for the multiplicative identity.
1. von Neumann regular element : An elementr of a ringR isvon Neumann regular if there exists an elementx ofR such thatr =rxr.
2. Avon Neumann regular ring: A ring for which each elementa can be expressed asa =axa for another elementx in the ring. Semisimple rings are von Neumann regular.
Azero ring: The ring consisting only of a single element0 = 1, also called thetrivial ring. Sometimes "zero ring" is used in an alternative sense to meanrng of square zero.
