  Let $\Lambda$ be a commutative Noetherian ring, and let $I$ be a proper ideal
of $\Lambda$, $R=\Lambda /I$. Consider the polynomial rings $T=\Lambda
[x_1,...x_n]$ and $A=R[x_1,...,x_n]$. Suppose that linear equations are
solvable in $\Lambda$. It is shown that linear equations are solvable in $R$
(thereby theoretically Gr\"obner bases for ideals of $A$ are well defined and
constructible) and that practically Gr\"obner bases in $A$ with respect to any
given monomial ordering can be obtained by constructing Gr\"obner bases in $T$,
and moreover, all basic applications of a Gr\"obner basis at the level of $A$
can be realized by a Gr\"obner basis at the level of $T$. Typical applications
of this result are demonstrated respectively in the cases where $\Lambda=D$ is
a PID, $\Lambda =D[y_1,...,y_m]$ is a polynomial ring over a PID $D$, and
$\Lambda =K[y_1,...,y_m]$ is a polynomial ring over a field $K$.
