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Let $\Lambda$ be a commutative Noetherian ring, and let $I$ be a proper idealof $\Lambda$, $R=\Lambda /I$. Consider the polynomial rings $T=\Lambda[x_1,...x_n]$ and $A=R[x_1,...,x_n]$. Suppose that linear equations aresolvable in $\Lambda$. It is shown that linear equations are solvable in $R$(thereby theoretically Gr\"obner bases for ideals of $A$ are well defined andconstructible) and that practically Gr\"obner bases in $A$ with respect to anygiven 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 applicationsof this result are demonstrated respectively in the cases where $\Lambda=D$ isa 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 $K$.

On the Construction of Gr\"obner Bases with Coefficients in Quotient Rings