On the Construction of Gröbner Bases with Coefficients in Quotient Rings

Abstract

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