Alternative letter-case form of Abelian algebra [(mathematics) Any algebra in which multiplication (or its equivalent) is commutative.]

(mathematics) A member of a self-dual topological ring built on the field of rational numbers (or, more generally, any algebraic number field), and involving in a symmetric way all the completions of the field.

(mathematics) A type of higher-order tensor used principally by Jan Arnoldus Schouten

(mathematics) A knot invariant which assigns a polynomial with integer coefficients to each knot type.

(countable, algebra) An algebraic structure consisting of a module over a commutative ring (or a vector space over a field) along with an additional binary operation that is bilinear over module (or vector) addition and scalar multiplication.

(algebra) A module (over some ring) with an additional binary operation, a module-element-valued product between module elements, which is bilinear over module addition and scalar multiplication. (N.B.: such bilinearity implies distributivity of the module multiplication with respect to the module addition, which means that such a module is also a ring.)

(algebra, field theory) A field extension L/K which is algebraic over K (i.e., is such that every element of L is a root of some (nonzero) polynomial with coefficients in K).

(algebra, logic) ring sum normal form

(mathematics, algebraic number theory) A field which includes the rational numbers and has finite dimension as a vector space over the rational numbers.

(algebra, of a ring) in which any descending chain of ideals eventually starts repeating.

(algebra) One of a pair of elements of an integral domain (or a ring) such that the two elements are divisible by each other (or, equivalently, such that each one can be expressed as the product of the other with a unit).

(mathematics) A theorem in set-theoretic geometry, which states that given a solid ball in three‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.

(mathematics) The prime number 1,000,000,000,000,066,600,000,000,000,001 which contains 666 (the Number of the Beast) and also two sets of thirteen zeroes.

(graph theory) A special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.

(number theory) A theorem stating that there are infinitely many prime numbers of the form a² + b⁴.

(algebra) A ring whose multiplicative operation is idempotent.

(computing theory) An algorithm for transforming a given set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order.

(algebra, ring theory) A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K.

(algebra, field theory, ring theory) For a given field or ring, a natural number that is either the smallest positive number n such that n instances of the multiplicative identity (1) summed together yield the additive identity (0) or, if no such number exists, the number 0.

(ring theory) A binary map in a given ring R, given by [a, b] = ab − ba, where a and b are elements of R, which yields the ring's zero element if and only if the multiplication operation commutes for a and b.

The number of multiplicands needed, at a minimum, to express a given group element as a product of commutators.

(mathematics) An ideal of a ring that measures how far it is from being integrally closed

(mathematics) The dual of a ring.

(group theory) A group generated by a single element.

(mathematics) A formula that connects trigonometry and complex numbers, stating that, for any complex number (and, in particular, for any real number) x and integer n, big ( cos (x)+i sin (x) big )ⁿ= cos (nx)+i sin (nx), where i is the imaginary unit.

(algebra, order theory) A bounded distributive lattice equipped with an involution (typically denoted ¬ or ~) which satisfies De Morgan's laws.

(algebra, ring theory) An integral domain in which every proper ideal factors into a product of prime ideals which is unique (up to permutations).

Dedekind domain

(ring theory) Such a set of tuples formed from two or more rings, forming another ring whose operations arise from the component-wise application of the corresponding original ring operations.

(algebra) An algebra over a field such that every non-zero element of it has a multiplicative inverse. (It is not required to have a unity element.)

(algebra) A ring with 0 ≠ 1, such that every non-zero element a has a multiplicative inverse, meaning an element x with ax = xa = 1.

(mathematics, logic) A kind of term in natural deduction.

(algebra) A complex number of the form a+b𝜔, where a and b are integers and ω is defined by the following two rules: (1) 𝜔³=1 and (2) 1+𝜔+𝜔²=0; an element of the Euclidean domain ℤ[𝜔].

(number theory) A conjecture about the distribution of prime numbers in arithmetic progressions, having many applications in sieve theory.

(mathematics) Of a ring: such that the ring and its residue field have the same characteristic.

(algebra) an integral domain in which division with remainder is possible

(algebra, field theory) A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements.

(algebra, ring theory) For given ring R, any ideal I such that Q = R / I, the set of cosets of elements of I in R, is a ring (the quotient ring of I in R).

(algebra, ring theory) A quotient ring.

(mathematics) A formula expressing the sum of the pth powers of the first n positive integers as a (p + 1)th-degree polynomial function of n, the coefficients involving Bernoulli numbers.

(algebra, ring theory) The smallest field in which a given ring can be embedded.

(algebra) A field all of whose elements can be represented as ordered pairs each of whose components belong to a given integral domain, such that the second component is non-zero, and so that the additive operator is defined like so: (a,b)+(a',b')=(ab'+a'b,bb'), the multiplicative operator is defined coordinate-wise, the zero is (0,1), the unity is (1,1), the additive inverse of (a,b) is (-a,b), equivalence is defined like so: (a,b)≡(a',b') if and only if ab'=a'b, and multiplicative inverse of a non-zero–equivalent element (a,b) is (b,a).

(singular only, mathematics) A particular lemma that claims the existence of a particular isomorphism in a commutative diagram given certain other homomorphisms in the diagram.

(algebra, ring theory) Given an integral domain R and its field of fractions K = Frac(R), an R-submodule I of K such that for some nonzero r∈R, rI ⊆ R.

(algebra, commutative algebra, field theory) Given a commutative ring R with prime characteristic p, the endomorphism that maps x → xᵖ for all x ∈ R.

(algebra, Galois theory) The automorphism group of a Galois extension.

(mathematics) A conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The first term in every such sequence appears to be 1.

A mathematical theorem, important for quantum logic, proving that the Born rule for the probability of obtaining specific results for a given measurement follows naturally from the structure formed by the lattice of events in a real or complex Hilbert space.

(mathematics) A theorem stating that every Goodstein sequence eventually terminates at zero.

(algebra) Given ring R with identity not equal to zero, and group G=g_1,g_2,...,g_n, the group ring RG has elements of the form a_1g_1+a_2g_2+...+a_ng_n (where a_i isin R) such that the sum of a_1g_1+a_2g_2+...+a_ng_n and b_1g_1+b_2g_2+...+b_ng_n is (a_1+b_1)g_1+(a_2+b_2)g_2+...+(a_n+b_n)g_n and the product is ∑ₖ₌₁ⁿ(∑_(g_ig_j=g_k)a_ib_j)g_k.

(computing theory) A particular kind of generating set of an ideal in a polynomial ring K[x1, ..,xn] over a field K. It allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite.

(mathematics) In real algebraic geometry, a theorem that describes the possible numbers of connected components that an algebraic curve can have, in terms of the degree of the curve.

(mathematics) A theorem stating that a polynomial ring over a Noetherian ring is Noetherian.

(mathematics) A basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism.

(algebra, Lie theory) A Lie subalgebra (subspace that is closed under the Lie bracket) 𝖍 of a given Lie algebra 𝖌 such that the Lie bracket [𝖌,𝖍] is a subset of 𝖍.

(algebra) For a ring R and an ideal m of R, the set of all u ∈ Frac(R) such that um ⊂ m, where Frac(R) is the fraction field of R.

(mathematics) An invertible element of the adele ring.

(algebra) An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element.

(algebra, ring theory) Any nonzero commutative ring in which the product of nonzero elements is nonzero.

(algebra, commutative algebra, ring theory) Given a commutative unital ring R with extension ring S (i.e., that is a subring of S), any element s ∈ S that is a root of some monic polynomial with coefficients in R.

(mathematics) The mean of two consecutive odd primes

(algebra, representation theory) The branch of algebra concerned with actions of groups on algebraic varieties from the point of view of their effect on functions.

(mathematics) a pair of integers (i,j) is an inversion pair of some permutation 𝜎 if i𝜎(j).

(algebra, of an element of a ring) that its only divisors are units and associates.

(mathematics) A non-associative algebra over a field whose product satisfies the axioms (i) xy=yx (the commutative law) and (ii) (xy)(xx)=x(y(xx)) (the Jordan identity).

(algebra) An algebra over a field; a ring with identity together with an injective ring homomorphism from a field, k, to the ring such that the image of the field is a subset of the center of the ring and such that the image of the field’s unity is the ring’s unity.

(mathematics) A basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero.

(mathematics, functional analysis) A proposition about convex sets in topological vector spaces.

(algebra) A ring obtained by adjoining a non-real complex pᵗʰ root of unity (where p is a prime number) to the ring of integers.

(mathematics) A formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X.

(algebra) A subring which is closed under left-multiplication by any element of the ring.

(algebra) An element of a structure that produces an identity mapping when applied on the left.

(slang or programming) Abbreviation of linear algebra. [(mathematics) The branch of mathematics that deals with vectors, vector spaces, linear transformations and systems of linear equations.]

(algebra) A commutative ring with a unique maximal ideal, or a noncommutative ring with a unique maximal left ideal or (equivalently) a unique maximal right ideal.

(algebra) A ring of fractions of a given ring, such that the complement of the set of allowed denominators is an ideal.

(algebra, ring theory) A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal.

(linear algebra) For a given square matrix M over some field K, the smallest-degree monic polynomial over K which, when applied to M, yields the zero matrix.

(mathematics) A complex analytic function satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.

(algebra) The largest sporadic group, of order 2⁴⁶ · 3²⁰ · 5⁹ · 7⁶ · 11² · 13³ · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 (approximately 8 · 10⁵³), denoted M or F₁.

(mathematics) A theorem used in complex analysis to establish a bound on the growth rates for an analytic function.

(mathematics) An algebraic structure similar to a ring but satisfying fewer axioms. They arise naturally from functions on groups.

Alternative form of near-ring (“algebraic structure”) [(mathematics) An algebraic structure similar to a ring but satisfying fewer axioms. They arise naturally from functions on groups.]

(algebra) The set of nilpotent elements of an algebraic structure such as an ideal

(algebra) Of a ring in which any ascending chain of ideals eventually starts repeating.

A Noetherian ring which is also an integral domain.

(algebra, ring theory) A ring which is either: (a) a commutative ring in which every ideal is finitely generated, or (b) a noncommutative ring that is both left-Noetherian (every left ideal is finitely generated) and right-Noetherian (every right ideal is finitely generated).

(algebra) A finite, algebraic extension of some field such that if it contains any root of any irreducible polynomial over that base field then it contains all of the roots of that polynomial.

(group theory) A subgroup H of a group G that is invariant under conjugation; that is, for all elements h of H and for all elements g in G, the element ghg⁻¹ is in H.

(mathematics) The neutral element with respect to multiplication in a ring.

(algebra) A theorem which states that for each element of a given set that a given group acts on, there is a natural bijection between the orbit of that element and the cosets of the stabilizer subgroup with respect to that element.

(algebra, order theory, ring theory) A ring, R, equipped with a partial order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.

(mathematics) An overring B of an integral domain A is a subring of the field of fractions K of A that contains A, i.e. A⊆B⊆K.

(algebra, field theory) A field K such that every irreducible polynomial over K has distinct roots.

(graph theory) The theorem stating that every cubic bridgeless graph contains a perfect matching.

(mathematics) A result giving an explicit description of the universal enveloping algebra of a Lie algebra.

(mathematics) A general-purpose integer factorization algorithm, particularly effective at splitting composite numbers with small factors.

(algebra) polynomial ring

(algebra, ring theory) A basis of a polynomial ring (said ring being viewed either as a vector space over the field of coefficients or as a free module over the ring of coefficients).

(algebra) A ring (which is also a commutative algebra), denoted K[X], formed from the set of polynomials (usually of one variable, in a given set, X), with coefficients in a given ring (often a field), K.

(mathematics) Given a group G that acts on a set P, a subset P is prefundamental for G if, for any g in G, gB is in B only if g is 1.

(algebra, ring theory) Given a commutative ring R, any ideal I such that for any a,b ∈ R, if ab ∈ I then either b ∈ I or aⁿ ∈ I for some integer n > 0.

In a commutative ring, a (two-sided) ideal I such that for arbitrary ring elements a and b, ab∈I⟹a∈I or b∈I.

(algebra, ring theory) Any nonzero ring R such that for any two (two-sided) ideals P and Q in R, the product PQ = 0 (the zero ideal) if and only if P = 0 or Q = 0.

(algebra, lattice theory, of a lattice) An element that is not a positive integer multiple of another element of the lattice.

(algebra, ring theory) A polynomial over an integral domain R such that no noninvertible element of R divides all its coefficients at once; (more specifically) a polynomial over a GCD domain R such that the greatest common divisor of its coefficients equals 1.

(algebra) An ideal I in an algebraic object R (which could be a ring, algebra, semigroup or lattice) that is generated by a given single element a ∈ R; the smallest ideal that contains a.

(algebra) An integral domain in which every ideal is a principal ideal.

(algebra, ring theory) A commutative ring in which every ideal is a principal ideal.

(mathematics, ring theory) A ring that is the direct product of rings.

(mathematics) A formula used to find the derivatives of products of two or more functions: (f·g)'=f'·g+f·g',!

(mathematics) The problem of finding two disjoint sets A and B of n integers each, such that ∑_(a∈A)aⁱ=∑_(b∈B)bⁱ for each integer i from 1 to a given k.

(mathematics) The set of all lower bounds of the set of all upper bounds of a subset of a partially ordered set

(mathematics) The intersection of the set of nonzero prime ideals of a ring.

(mathematics) Any of various variants of a ring.

(algebraic number theory) A number field that is an extension field of degree two over the rational numbers.

(algebra, ring theory) For a given ring R and ideal I contained in R, another ring, denoted R / I, whose elements are the cosets of I in R.

(algebra) The monoid action of a ring R on an abelian group.

(algebra, ring theory, of a module) The intersection of maximal submodules of a given module.

(algebra, commutative algebra, ring theory) An ideal I within a ring R that is its own radical (i.e., for any r ∈ R, if rⁿ ∈ I for some positive integer n, then r ∈ I).

(algebra) The maximum quantity of D-linearly independent elements of a module (over an integral domain D).

(algebra, ring theory) An algebra over a reduced ring.

(algebra, ring theory) A ring R that has no nonzero nilpotent elements; equivalently, such that, for x ∈ R, x² = 0 implies x = 0.

(algebra) An element of a ring that is not a zero divisor.

(group theory) An expression of the identity element of a group as a product of generators, used in a presentation (type of specification) of the group.

(number theory) A coset of an ideal of the ring of integers.

(algebra) A copy of a subgroup, multiplied on the right by some element from the parent group.

(algebra) A subring which is closed under right-multiplication by any element of the ring.

(algebra) An element of a structure that produces an identity mapping when applied on the right.

(mathematical analysis, measure theory) A family of sets that is closed under finite unions and differences.

(algebra) A ring whose elements are fractions whose numerators belong to a given commutative unital ring and whose denominators belong to a multiplicatively closed unital subset D of that given ring. Addition and multiplication of such fractions is defined just as for a field of fractions. A pair of fractions a/b and c/d are deemed equivalent if there is a member x of D such that x(ad-bc)=0.

An algebraist who specializes in ring theory

(algebra) An algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element.

(algebra, Galois theory) splitting field

(abstract algebra) multiplication of a module element by a ring element

(mathematics) An algebraic structure with two binary operations, addition and multiplication, similar to the field but with some axioms relaxed.

(mathematics) Describing a ring that has a finite number of maximal ideals

(mathematics) A polynomial sequence in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics.

(computing theory) A quantum algorithm for finding the prime factors of an integer.

(mathematics) A type of potential counterexample to the generalized Riemann hypothesis, on the zeros of Dirichlet L-function.

(algebra, field theory) A field extension obtained by adjoining a single element to a given field.

(algebra, ring theory) A ring that contains no nontrivial ideals (i.e., no (two-sided) ideals other than the zero ideal and the ring itself).

(algebra, dated) A ring in which every nonzero element has a multiplicative inverse; division ring

An algorithm for computing the value of a transcendental number, generating its digits sequentially with increasing precision.

(algebra, Galois theory) (of a polynomial) Given a polynomial p over a field K, the smallest extension field L of K such that p, as a polynomial over L, decomposes into linear factors (polynomials of degree 1); (of a set of polynomials) given a set P of polynomials over K, the smallest extension field of K over which every polynomial in P decomposes into linear factors.

(mathematics) In combinatorial mathematics, a type of block design, specifically a t-design with λ = 1 and t = 2 or (recently) t ≥ 2.

(algebra) A subring of a field, containing the multiplicative identity and closed under inversion.

(algebra) A module contained in a larger module, both over the same ring, such that the ring multiplication in the former is a restriction of that in the latter.

(algebra) a ring which is contained in a larger ring, such that the multiplication and addition on the former are a restriction of those on the latter.

(algebraic geometry) A theorem stating that a normal algebraic variety with an action of a torus can be covered by torus-invariant affine open subsets.

(mathematics) Any of a collection of theorems of finite group theory that give detailed information about the number of subgroups of fixed order that a given finite group contains.

(mathematical analysis) A truncated Taylor series; the sum of the first n terms of a Taylor series.

(algebra) A ring of fractions of a given ring, such that the set of allowed denominators contains all of the non-zero divisors of the given ring.

(linear algebra) A matrix (of dimension n×m) that represents some linear transformation from ℝᵐ→ℝⁿ.

(algebra) A subring which is both a left ideal and a right ideal. Commonly referred to simply as an ideal, but referred to as two-sided ideal for emphasis.

(algebra, ring theory) A unique factorization ring which is also an integral domain.

(algebra, ring theory) A ring in which every non-zero, non-unit (i.e., proper) element can be factored into a product of prime elements (not necessarily distinct), such that such factorization is unique up to permutation of the factors and the replacing of primes by their associate primes.

(algebra) An element having an inverse, an invertible element; an associate of the unity.

(algebra) An integral domain D such that for every element x of its field of fractions F, at least one of x or x⁻¹ belongs to D.

(algebraic geometry) An ideal consisting of all polynomials (belonging to a certain polynomial ring) which zero out within a given algebraic variety.

(algebra) The ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form f_m(X)∂_Xᵐ+f_m-1(X)∂_Xᵐ⁻¹+⋯+f_1(X)∂_X+f_0(X).

A theorem stating that two connected graphs are isomorphic if and only if their line graphs are isomorphic, with a single exception: K₃, the complete graph on three vertices, and the complete bipartite graph K_(1,3), which are not isomorphic but both have K₃ as their line graph.

(algebra, ring theory) An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.