algebra over a field

algebra over a field (plural algebras over a field)

1. (algebra) A vector space (over some field) with an additional binary operation, a vector-valued product between vectors, which is bilinear over vector addition and scalar multiplication. (N.B.: such bilinearity implies distributivity of the vector multiplication with respect to the vector addition, which means that such a vector space is also a ring.)

• algebra
• k-algebra

• ring
• vector space

• associative algebra
• division algebra
• non-associative algebra

• algebra over a ring