Algebra over a field

An algebra over a field is a vector space equipped with a bilinear multiplication. The multiplication may also satisfy conditions such as associativity or having an identity element.

Vector space: A set with addition and scalar multiplication satisfying axioms that support linear combinations. An algebra is first a vector space, with multiplication added.
Associative algebra: An algebra whose multiplication satisfies (ab)c = a(bc) for all elements. Associativity is one common extra law imposed on an algebra.
Matrix algebra: An algebra of matrices with matrix addition, scalar multiplication, and matrix multiplication. Matrices provide a standard concrete example of an associative algebra.
Ring: A set with addition and multiplication satisfying ring axioms, without requiring scalar multiplication by a field. A ring need not carry the vector-space structure required of an algebra over a field.
Field (mathematics): A commutative ring in which every nonzero element has a multiplicative inverse. Its elements provide the scalars used to define the algebra.
Unital algebra: An algebra with a multiplicative identity element. An identity provides a neutral element for the multiplication.
Polynomial algebra: An algebra whose elements are polynomials with coefficients in a specified field. Polynomial multiplication gives a basic commutative algebra used throughout mathematics.
Semigroup: A set equipped with an associative binary operation. A semigroup has multiplication but lacks the linear structure and bilinearity of an algebra.
Bilinear map: A function of two vector inputs that is linear in each input when the other is held fixed. Bilinearity is the compatibility rule for the algebra's multiplication.
Commutative algebra: An associative algebra whose multiplication satisfies ab = ba for all elements. Commutativity is an additional restriction on multiplication.

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Representation theoryRelated: Representations of algebras preserve both vector-space structure and algebra multiplication.