From wikipedia:
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, that satisfy the axioms implied by "vector space"
and "bilinear".[1]
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
that give both structures and names equal priority.
Wikipedia
