Let $k$ be a commutative ring and let $A=k\langle X_1,...,X_n\rangle/(f_i\mid i\in I)$, $B=k\langle Y_1,...,Y_m\rangle/(g_j\mid j\in J)$ be quotients of free noncommutative $k$-algebras.
Question: Is it true, that the free product of $A$ and $B$ is given by $$A\ast B=k\langle X_1,...,X_n,Y_1,...,Y_m\rangle/(f_i,g_j\mid i\in I, j\in J)$$