TITLE: Coproduct in the category of (noncommutative) associative algebras
QUESTION [18 upvotes]: 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.

REPLY [5 votes]: In the book Rings with generalized identities by Beidar, Martindale and Mikhalev, Section 1.4 (from page 24 onwards) you can find the definition from commutative diagrams, the characterization with proof (which is the one given by Chindea Filip in his answer) and several properties.
The main purpose of the coproduct in this book is to serve as the "home" for generalized polynomials identities for prime and semiprime rings, which are, informally speaking, like polynomial identities in which in addition it is allowed to insert elements from the ring (or even from any of its usual rings of quotients). So, for example $xy-yx$ is a PI, while $pxy-yqx$ with $p,q$ fixed elements of the maximal ring of quotients is a GPI.
The mentioned part of the book is readable through Google Books:
Rings with generalized identities - Coproducts
and the book is downloadable elsewhere.