Pushout of unital non commutative algebras I like to know if there is a pushout in the category of non commutative alegbras with unit and if the answer is "yes", who is it?

Given unital $R$-algebras $A\leftarrow B\to C$, the pushout $A \star_B C$ is generated as an $R$-algebra by generators of $A$ and of $C$, modulo the union of the relations in $A$ and in $C$, as well as further relations identifying the two resulting images of each element of $B$. This immediately gives the canonical maps from $A$ and $C$. You can describe this construction as a quotient of the free $R$-module on words with letters from $A$ and $C$, if you like. 
