Given an associative algebra $A$, with enveloping algebra $A^e = A\otimes A^{op}$, the Takeuchi product $\times_A$ is certain product in the category of $A$-rings. It generalizes a construction of M. E. Sweedler where $A$ 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.

- Mitsuhiro Takeuchi,
*Groups of algebras over $A \times \bar{A}$, J. Math. Soc. Japan***29**, 459–492, 1977, MR0506407, euclid - M. E. Sweedler,
*Groups of simple algebras*, Publ. IHES**44**, 79–189, MR51:587, numdam - T. Brzeziński, G. Militaru,
*Bialgebroids, $\times_{R}$-bialgebras and duality*, J. Algebra 251: 279-294, 2002, math.QA/0012164 - P. Schauenburg,
*Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules*, Appl. Categ. Structures**6**(1998), 193–222, ps doi

category: algebra

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