Algebra¶

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28 domain-specific abstractions whose origin domain is Algebra.

Absolute value (algebra) — A nonnegative multiplicative or submultiplicative magnitude function on a field or integral domain that separates zero and satisfies the triangle inequality.
Absorbing element — An element that returns itself whenever combined with any element under a specified binary operation.
Anticommutative property — A binary-operation property in which exchanging the two arguments yields the additive inverse of the original result.
Binomial (polynomial) — A polynomial consisting of exactly two nonzero monomial terms, whose sparse form supports binomial expansions, toric ideals and algebraic varieties governed by exponent differences.
Cancellation property — An algebraic property allowing a common left or right factor to be removed from an equality, even when no inverse element is available.
Commutator — An algebraic expression that measures failure of two elements or operators to commute, such as aba⁻¹b⁻¹ in a group or ab−ba in a ring.
Cubic function — A polynomial function of degree exactly three with nonzero leading coefficient.
Cyclotomic polynomial — The monic irreducible integer polynomial whose roots are exactly the primitive nth roots of unity.
Diagonal form — A homogeneous polynomial containing only pure powers of individual variables and no mixed monomials.
Filtered algebra — An algebra equipped with a nested exhaustive sequence of subspaces whose multiplication sends filtration levels p and q into level p+q.
Formally real field — A field that admits an ordering compatible with its operations, equivalently one in which minus one cannot be expressed as a finite sum of squares.
Gerstenhaber algebra — A graded-commutative algebra equipped with a degree-minus-one graded Lie bracket that acts as a graded derivation of the product.
Grothendieck group — The universal abelian group completion of a commutative monoid, formally adjoining additive inverses while preserving every monoid homomorphism into an abelian group.
Homomorphism — A map between algebraic structures of the same signature that preserves each distinguished operation and constant.
Hurwitz problem — The problem of determining when sums-of-squares quadratic forms admit bilinear multiplicative composition formulas.
J-structure — An algebraic structure taking a rational inversion map and Hua-type identities as primitive, providing a linear-algebraic-group formulation closely equivalent to Jordan algebra theory in suitable characteristic.
Lie coalgebra — A vector space with a skew-symmetric cobracket satisfying the co-Jacobi identity, dual to a Lie algebra in finite dimensions.
Locally nilpotent — A local finiteness condition under which every finitely generated subobject is nilpotent, or an ideal becomes nilpotent after localization at a specified prime.
Necklace ring — A ring on infinite sequences over a commutative ring whose multiplication combines indices by least common multiple and weights products by greatest common divisor.
Ordered field — A field with a total order preserved by addition and multiplication by positive elements.
Quadratic function — A polynomial function of degree exactly two, represented by a nonzero quadratic form plus lower-degree terms.
Quintic function — A polynomial function of degree exactly five with nonzero leading coefficient.
Real Closed Field — Identify an ordered field maximal among orderable algebraic extensions, equivalently one where positive elements are squares and every odd-degree polynomial has a root.
Semiprimitive ring — A ring with zero Jacobson radical, equivalently one whose simple modules collectively detect every nonzero element.
Sparse polynomial — A polynomial represented by relatively few nonzero monomial terms compared with its degree, dimension or dense coefficient array.
Symmetric polynomial — A multivariable polynomial unchanged by every permutation of its variables.
Total algebra — An algebra of all coefficient functions on a suitably finite-factorization monoid, with convolution multiplication extending the finite-support monoid algebra to infinite formal sums.
Uniform module — A nonzero module in which every two nonzero submodules intersect nontrivially, equivalently every nonzero submodule is essential.