In mathematics, an algebra over a field is a vector space equipped with a bilinear product. An algebra such that the product is associative and has an identity is therefore a ring that is also a vector space, and thus equipped with a field of scalars. Such an algebra is called here a unital associative algebra for clarity, because there are also nonassociative algebras.
In other words, an algebra over a field is a set together with operations of multiplication, addition, and scalar multiplication by elements of the underlying field, which satisfy the axioms implied by "vector space" and "bilinear".
One may generalize this notion by replacing the field of scalars by a commutative ring, and thus defining an algebra over a ring.
Because of the ubiquity of associative algebras, and because many textbooks teach more associative algebra than nonassociative algebra, it is common for authors to use the term algebra to mean associative algebra. However, this does not diminish the importance of nonassociative algebras, and there are texts which give both structures and names equal priority.
Read more about Algebra Over A Field: Kinds of Algebras and Examples, Algebras and Rings, Structure Coefficients, Classification of Low-dimensional Algebras
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