coproduct of noncommutative algebra and commutative algebras

Question

I have read the book "Rings with generalized identities" and I understand that the free product of asociative unital algebras are the coproduct of them, but I can't understand why this reduces to the tensor product of them when alegbras are conmutative.

Answer 1

The original question seems to be missing an important subtlety: the inclusion of a subcategory into a category need not preserve coproducts.

The coproduct of two commutative algebras is two different things depending on whether you take the coproduct in the category of commutative algebras or noncommutative algebras:

• The coproduct in the category of commutative algebras is the tensor product.
• The coproduct of commutative algebras, in the category of noncommutative algebras, is still the free product.