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It is known that the tensor product of two commutative algebras is their coproduct. It has to do with well-definedness of the multiplication for the homomorphism appearing from the universal property (it is very well explained in Proof that the tensor product is the coproduct in the category of R-algebras).

On the other hand, there is a very good explanation about which the form of the elements into a general coproduct (not only for the commutative case) is in How to construct the coproduct of two (non-commutative) rings (extending to algebras). It takes a relation that makes the units to disappear all along the products.

My question is, since the tensor product of non-commutative algebras depends on a universal property and makes use of the commutator (according to https://en.wikipedia.org/wiki/Tensor_product_of_algebras): how that tensor can be constructed? Which are its elements and multiplication?

gibarian
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    It's defined in the definition part of the wiki page. In particular, it's just the usual tensor product of modules, with the componentwise multiplication. – jgon Oct 14 '20 at 17:52
  • That's the point I wanted to reach: the multiplication for the quotient described nicely in: math.stackexchange.com/questions/625874/… does not satisfy the componentwise multiplication requirement. In fact, that description looks like the tensor algebra rather than the tensor product of algebras. I'm confused. – gibarian Oct 15 '20 at 07:45

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