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

Let $\mathrm{\Lambda }$ be a commutative Noetherian ring, and let $I$ be a proper ideal of $\mathrm{\Lambda }$, $R=\mathrm{\Lambda }\mathrm{/}I$. Consider the polynomial rings $T=\Lambda [x_1,…x_n]$and$A=R[x_1,…,x_n]$. Suppose that linear equations are solvable in $\mathrm{\Lambda }$. 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 $\mathrm{\Lambda }=D$ is a PID, $\mathrm{\Lambda }=D[y_1,\mathrm{.}\mathrm{.}\mathrm{.},y_m]$ is a polynomial ring over a PID $D$, and $\mathrm{\Lambda }=K[y_1,\mathrm{.}\mathrm{.}\mathrm{.},y_m]$ is a polynomial ring over a field $K$.