This page gathers the 289 dictionary entries belonging to the domain “Algebra”, as labelled by Wiktionary. Each word leads to its full entry: definitions, etymology, pronunciation, examples. Page 2 of 3: from “Euclid's lemma” to “p-adic norm”.
- Euclid's lemma noun the proposition that for elements a, b, c of a given principal ideal domain, if a divides bc and gcd(a, b) = 1, then a divides c.
- exact sequence noun A sequence of modules (or groups, vector spaces, etc.; most generally, objects of an abelian category) with adjacent objects connected by…
- expand verb To rewrite (an expression) as a longer, yet equivalent, sum of terms.
- expansion noun The rewriting of an expression as a longer but equivalent sum of terms.
- extension field noun A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements.
- exterior product noun A kind of product between vectors and/or multivectors which is associative, linear, and alternating.
- factor ideal noun 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).
- factor out verb To isolate a common factor from an expression.
- field noun A non-zero commutative ring in which all non-zero elements are invertible; a simple commutative ring.
- field of quotients noun 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…
- finitely generated adj Such that the functor operatorname Hom_( mathcal )C(X,·) preserves those filtered colimits of monomorphisms.
- finitely presented adj Isomorphic to the kernel of a morphism of free modules; such that there exists an exact sequence Rᵐ→Rⁿ→M→0 for some integers m,n.
- fixed field noun A subfield of a given field which contains all of the fixed points that are common to all of the automorphisms of some subgroup of the…
- flat adj Such that its target, regarded as a module over its source, is flat (as above).
- formal power series noun Any finite or infinite series of the form a_0+a_1x+a_2x²…=∑ᵢa_ixⁱ, where the aᵢ are numbers, but it is understood that no value is assigned…
- formal sum noun An element of a free abelian group or vector space regarded as a sum of basis elements.
- fractional ideal noun 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.
- free adj Left adjoint to a forgetful functor G; such that any map f:X→G(A) induces a universal map ̄f:F(X)→A.
- free Boolean algebra noun A field of sets whose elements are equivalent to Boolean formulas (or, perhaps more precisely, equivalence classes of Boolean formulas).…
- free module noun A module that has a basis. Equivalently, a module consisting of n-tuples of ring elements with no extra identities. (Then the number n is…
- free monoid noun A monoid whose underlying set is the Kleene closure of some set of generators, and whose operator is concatenation.
- Frobenius endomorphism noun Given a commutative ring R with prime characteristic p, the endomorphism that maps x → xᵖ for all x ∈ R.
- Galois extension noun An algebraic extension that is both a normal and a separable extension; equivalently, an algebraic extension E/F such that the fixed field…
- Galois theory noun The branch of mathematics dealing with Galois groups, Galois fields, and polynomial equations. It provides a link between field theory and…
- Grassmann algebra noun A direct sum with an exterior product of multivector spaces which are all based on a same underlying finite-dimensional vector space. The…
- group ring noun 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…
- group theory noun The mathematical theory of groups.
- groupoid noun A magma: a set with a total binary operation.
- Heyting algebra noun A bounded lattice, L, modified to serve as a model for a logical calculus by being equipped with a binary operation called "implies"…
- homogeneous adj Such that all its nonzero terms have the same degree.
- homological algebra noun The branch of mathematics dealing with homology and in particular with homological functors and the algebraic structures they entail.
- homology noun Given a chain complex {Gₙ} and its associated set of homomorphisms {Hₙ}, the rule which explains how each Hₙ maps Gₙ into the kernel of…
- homomorphism noun A structure-preserving map between two algebraic structures of the same type, such as groups, rings, or vector spaces.
- Hurwitz algebra noun Any one of the unital composition algebras identified by Hurwitz's theorem (on composition algebras) as solutions to the Hurwitz problem.
- hyperbolic quaternion noun A nonassociative algebra over the real numbers with elements of the form: q = a + bi + cj + dk, a,b,c,d {R}
- ideal noun A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.
- idealizer noun 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.
- identity noun Any function which maps all elements of its domain to themselves.
- identity element noun An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other…
- independent variable noun In an equation, any variable whose value is not dependent on any other in the equation.
- indeterminate noun A symbol that resembles a variable or parameter but is used purely formally and neither signifies nor is ever assigned a particular value;
- index noun The number of cosets that exist.
- injective adj Loosely, having a certain generalizing property, abstracted from the study of ℚ as a ℤ-module. Formally, such that any short exact sequence…
- integral adj Being the root of some monic polynomial in A.
- integral element noun 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…
- invariant theory noun The branch of algebra concerned with actions of groups on algebraic varieties from the point of view of their effect on functions.
- inverse limit noun A subset of the Cartesian product of all the members of an inverse system, such that a member M of the subset is a sort of “cross section”…
- inverse system noun A set of algebraic structures (which are part of a concrete category; e.g., groups) and a set of morphisms between them (e.g., group…
- inversion noun An operation on a group, analogous to negation.
- irreducible adj Whose only divisors are units and associates.
- isolated adj Minimal with respect to inclusion (among associated primes).
- join noun The lowest upper bound, an operation between pairs of elements in a lattice, denoted by the symbol ∨.
- k-algebra noun 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…
- K-theory noun The study of rings R generated by the set of vector bundles over some topological space or scheme;
- kernel noun The set of elements of G which are mapped to the identity of H (usually written as 0 in an additive group and as 1 in a multiplicative…
- Kleene algebra noun A De Morgan algebra which also satisfies the inequation x∧∼x⩽y,∨∼y for all x and y, where "∼" here denotes the De Morgan involution.
- known noun A constant or variable the value of which is already determined.
- Kummer ring noun A ring obtained by adjoining a non-real complex pᵗʰ root of unity (where p is a prime number) to the ring of integers.
- Lagrange's theorem noun The theorem in group theory that for any finite group, the size (order) of any subgroup must be a divisor of that of the group itself.
- lattice noun A discrete subgroup of Rⁿ which is isomorphic to Zⁿ (considered as an additive group) and which spans the real vector space Rⁿ.
- left coset noun A left coset of a subgroup is a copy of that subgroup, multiplied on the left by some element from the parent group
- lie over verb Of a prime ideal q of a ring R' with subring R, to intersect with R at some other prime ideal p; q lies over p if and only if q∩R=p.
- linear algebra noun an associative algebra
- linear dependence noun The state of being linearly dependent
- local adj Such that the following conditions are equivalent: (1) P holds for R (M); (2) P holds for the localization R_p (M_p) for all prime ideals p…
- local ring noun A commutative ring with a unique maximal ideal, or a noncommutative ring with a unique maximal left ideal or (equivalently) a unique…
- localization noun A systematic method of adding multiplicative inverses to a ring.
- localize verb To produce (from a ring and an ideal in that ring) the ring of fractions, where the set of allowed denominators is the compliment of the…
- loop noun A quasigroup with an identity element.
- matrix algebra noun Generically, the arithmetic of matrices, involving matrix addition, scalar multiplication of matrices, matrix multiplication, and matrix…
- maximal ideal noun An ideal which cannot be made any larger (by adjoining any element to it) without making it improper (i.e., equal to the whole of the…
- maxterm noun A sum that is a logical OR of a set of variables where each individual variable only appears once in the sum, either in complemented or…
- meet noun The greatest lower bound, an operation between pairs of elements in a lattice, denoted by the symbol ∧.
- module noun An abelian group equipped with the operation of multiplication by an element of a ring (or another of certain algebraic objects)…
- monoid noun A set which is closed under an associative binary operation, and which contains an element which is an identity for the operation.
- multibracket noun A partition of a set into disjoint subsets such that each subset contains an even number of elements, and that if those elements are…
- multiplicative adj Having multiplication as an operator.
- multiplicative identity noun An element of an algebraic structure, generally denoted 1, which is an identity for a multiplicative operation (generally denoted × or *…
- natural adj Closed under submodules, direct sums, and injective hulls.
- nilpotent adj Such that, for some positive integer n, xⁿ = 0.
- noetherian adj In which any ascending chain of ideals eventually becomes constant.
- Noetherian module noun A module whose every submodule is finitely generated; or equivalently, a module whose submodules satisfy an ascending chain condition…
- Noetherian ring noun A ring which is either: (a) a commutative ring in which every ideal is finitely generated, or (b) a noncommutative ring that is both…
- non-associative algebra noun An algebra over a ring (or more narrowly, an algebra over a field) whose bilinear product is not necessarily associative.
- nonassociative adj Whose multiplication operation is not assumed to be associative for all elements.
- nonpolynomial adj Not polynomial.
- norm noun An element of the image of some (generalized) norm, the element then said to be from the norm in question, or from the structure which gave…
- normal adj With cosets which form a group.
- normal basis noun For a given Galois field 𝔽_(qᵐ) and a suitable element β, a basis that has the form {β, β^q, β^(q2), ... , β^(qm-1)}.
- normal extension noun A finite, algebraic extension of some field such that if it contains any root of any irreducible polynomial over that base field then it…
- normalizer noun The subset of elements of some group which leave some given subset invariant when conjugating it.
- normed adj Of a mathematical structure, endowed with a norm.
- Ockham algebra noun A bounded distributive lattice with a dual endomorphism (where “dual” means that it satisfies De Morgan’s laws).
- order noun The sum of the exponents of the variables involved in the expression.
- ordered field noun A field which has an order relation satisfying these properties: trichotomy, transitivity, preservation of an inequality when the same…
- ordered integral domain noun An integral domain which has a subset whose elements are said to be "positive", such that this subset is closed under addition, closed…
- ordered ring noun 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…
- orthotomic adj imaginary; involving the square root of a negative number.
- osculant noun The condition that the solution to a set of simultaneous quantics, is also the solution of the corresponding set of tangential quantics.
- p-adic norm noun A norm on a vector space which is defined over a field equipped with a discrete valuation (a generalisation of p-adic absolute value).