Coproduct in the category of (noncommutative) associative algebras For the case of commutative algebras, I know that the coproduct is given by the tensor product, but how is the situation in the general case? (for associative, but not necessarily commutative algebras over a ring $A$). Does the coproduct even exist in general or if not, when does it exist? If it helps, we may assume that $A$ itself is commutative.
I guess the construction would be something akin to the construction of the free products of groups in group theory, but it would be nice to see some more explicit details (but maybe that would be very messy?) I did not have much luck in finding information about it on the web anyway.

Quillin and Cuntz desribe this in the book "Cyclic Homology in non-commutative geometry" on the first page, without proof.
The construction goes as follows:
$A\star B \cong A \oplus B \oplus (A \otimes B) \oplus (B \otimes A) \oplus (A \otimes A \otimes B) \oplus (A \otimes B \otimes A) \oplus (B \otimes A \otimes A) \oplus ...$ (and keeps cycling though tensors).
Its a construction known as the free product of associative algebras.  Though it should be no canonical embedding of A (or B) into any of the tensor products above, since any arrow $A \rightarrow A \otimes B$ cannot be free of relations.
