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Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".

Products, Kinds of algebras and examples & Overview

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Overview

Definition and motivation

Basic concepts

Kinds of algebras and examples

Algebras and rings

Structure coefficients

Classification of low-dimensional unital associative algebras over the complex numbers

Associative algebras over rings

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Map overview Semantic statistics

Algebra over a field

Nodes105
Edges104
Triples18
Avg. degree1.98
Density0.019048
Components1

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Algebra over a field

Top relations

related to Associative algebras over rings · 9
Algebra over a field → In, Note, On, R-algebra, R-module, See Field, So, The, Z-module
related to Algebras and rings · 5
Algebra over a field → In, K-algebra, Since, The, This
related to Subalgebras and ideals · 2
Algebra over a field → In, K-algebra

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Important terminology

algebra field ring multiplication algebras space unital associative commutative vector element structure product bilinear elements given example identity homomorphism basis

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
algebraic geometryinstance ofor in some subjects0.80text
unital associative commutative algebra.Replacing the field of scalars by a commutative ring leads to the more general notion of an algebra over a ringinstance ofor in some subjects0.80text
Algebra over a fieldrelated to Algebras and ringsThe0.60section
Algebra over a fieldrelated to Algebras and ringsK-algebra0.60section
Algebra over a fieldrelated to Algebras and ringsIn0.60section
Algebra over a fieldrelated to Algebras and ringsSince0.60section
Algebra over a fieldrelated to Algebras and ringsThis0.60section
Algebra over a fieldrelated to Associative algebras over ringsThe0.60section
Algebra over a fieldrelated to Associative algebras over ringsNote0.60section
Algebra over a fieldrelated to Associative algebras over ringsR-algebra0.60section
Algebra over a fieldrelated to Associative algebras over ringsIn0.60section
Algebra over a fieldrelated to Associative algebras over ringsR-module0.60section

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