relations of tensor product are relations of original rings?

Question

Given two (finitely-generated, not necessarily commutative) rings $A$ and $B$, given by generating sets $R$ and $S$, respectively, and relation sets $R^*$ and $S^*$, respectively. Is the tensor product $A \otimes B$ given by the generating set $\{r \otimes s \mid r \in R, s \in S\}$ and relations $\{a \otimes 1, 1 \otimes b \mid a \in R^*, b \in S^*\}$ as well as the tensor relations?