  Let $K\langle X\rangle =K\langle X_1,...,X_n\rangle$ be the free algebra of
$n$ generators over a field $K$, and let $R\langle X\rangle =R\langle
X_1,...,X_n\rangle$ be the free algebra of $n$ generators over an arbitrary
commutative ring $R$. In this semi-expository paper, it is clarified that any
monic Gr\"obner basis in $K\langle X\rangle$ may give rise to a monic Gr\"obner
basis of the same type in $R\langle X\rangle$, and vice versa. This fact turns
out that many important $R$-algebras have defining relations which form a monic
Gr\"obner basis, and consequently, such $R$-algebras may be studied via a nice
PBW structure theory as that developed for quotient algebras of $K\langle
X\rangle$ in ([LWZ], [Li2, 3]).
