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.

The following is a link to an article which provides a partial answer, namely it gives (on page 8, without proof) the coproduct of two non-commutative algebras (over a field rather than a ring, I don't know the ring case) http://www.google.co.uk/url?q=http://citeseerx.ist.psu.edu/viewdoc/download%3Fdoi%3D10.1.1.6.6129%26rep%3Drep1%26type%3Dpdf&sa=U&ei=PK3IUeLGIdHktQbUsoCwAQ&ved=0CB4QFjAB&usg=AFQjCNHZM3ux74AVdgFECW5HPfM3syw9rg
