Pushout of associative algebras How to describe a pushout in the category of associative algebras? I guess it should be something like an amalgamated free product of groups, but I have not found any reference.
I came up with the following construction. Let $f\colon R\to A$ and $g\colon R\to B$ be morphisms of algebras. We can form a free product $A*B$. Let $I$ be the ideal in $A*B$ generated by $f(r)-g(r)$ for $r\in R$. Then, the pushout should be $A*_R B=A*B/I$. Is the construction correct?
I am looking forward for some references on this topic. Particularly about the connection with the Cohn localization (is it true that pushout of a Cohn localization is a Cohn localization?)

Your construction is correct and works in a very general setting. If a category has coproducts and coequalizers, then it has pushouts. Namely, the pushout of $X \to Y$ and $X \to Z$ is the coequalizer of $X \to Y \to Y \sqcup Z$ and $X \to Z \to Y \sqcup Z$. This appears in every category theory textbook.
