Existence and Construction of a Gr\"obner Basis for a Polynomial Ideal

TL;DR

This paper presents a method to construct a Gr"obner basis for polynomial ideals over a commutative ring by leveraging the existence of Gr"obner bases in the base ring itself, extending the applicability of Gr"obner basis algorithms.

Contribution

It introduces a construction that lifts Gr"obner basis algorithms from the base ring to polynomial ideals over that ring, assuming the ring admits Gr"obner bases for all its ideals.

Findings

Provides a systematic way to lift Gr"obner bases from rings to polynomial ideals.

Extends the applicability of Gr"obner basis algorithms to broader algebraic structures.

Abstract

This extended abstract gives a construction for lifting a Gr\"obner basis algorithm for an ideal in a polynomial ring over a commutative ring R under the condition that R also admits a Gr\"obner basis for every ideal in R.