Given an associative algebra , with enveloping algebra , the Takeuchi product is a certain bifunctor in the category of -rings. It generalizes a construction of M. E. Sweedler where is commutative; Sweedler’s article may be itself viewed in a sense a “generalization of the relative Brauer group and the associated theory”.
Takeuchi product is used in the theory of associative bialgebroids over noncommutative base.