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

TL;DR

This paper demonstrates how to construct Gröbner bases over quotient rings by leveraging Gröbner bases in polynomial rings over Noetherian rings, enabling practical computation in these algebraic structures.

Contribution

It establishes that Gröbner bases in quotient polynomial rings can be effectively constructed via Gröbner bases in polynomial rings over Noetherian rings, broadening computational methods.

Findings

Linear equations are solvable in quotient rings if solvable in the base ring.

Gröbner bases in quotient rings can be obtained from those in polynomial rings over the base ring.

Applications include polynomial rings over PIDs and fields, demonstrating practical utility.

Abstract

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 $K$.