Associative Algebra

An algebra A over a commutative ring (often a field) that is simultaneously a module over that ring and a ring whose multiplication is associative and bilinear with respect to the scalar action; unital or nonunital variants occur.

A vector space or module over a commutative ring or field equipped with a bilinear multiplication map that is associative: (ab)c = a(bc) for all elements a,b,c. The algebra structure pairs the linear structure with a single associative binary product.