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( a + i b ) ⋅ ( c + i d ) is an algebra over F.Īlgebras over fields come in many different types.

6 Classification of low-dimensional unital associative algebras over the complex numbersĭefinition and motivation Motivating examples Algebra.Algebras are not to be confused with vector spaces equipped with a bilinear form, like inner product spaces, as, for such a space, the result of a product is not in the space, but rather in the field of coefficients. Replacing the field of scalars by a commutative ring leads to the more general notion of an algebra over a ring. Many authors use the term algebra to mean associative algebra, or unital associative algebra, or in some subjects such as algebraic geometry, unital associative commutative algebra. It is an example of a unital associative algebra, a (unital) ring that is also a vector space. The ring of real square matrices of order n forms a unital algebra since the identity matrix of order n is the identity element with respect to matrix multiplication. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity instead.Īn algebra is unital or unitary if it has an identity element with respect to the multiplication. Given an integer n, the ring of real square matrices of order n is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. 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". In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.