Commutative von Neumann regular rings are 1-Gröbner

Zoran Petrovic,Maja Roslavcev

doi:10.2298/aadm210419030p

Abstract

Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gr?bner basis. We prove this result using only the defining property of a von Neumann regular ring.

Highlights

When considering the Grobner bases for ideals in polynomial rings over a ring R, most of the theory and applications is developed for R — a field or a Noetherian commutative domain; see [1], [2]
A von Neumann regular ring is another kind of zero-dimensional ring and here we prove that it is 1-Grobner, if it is commutative
We give the basic definitions concerning the Grobner bases and present the important Lemma 9. This lemma enables us to prove the main result of the paper: every finitely generated ideal in R[X] has a strong Grobner basis

Summary

INTRODUCTION

When considering the Grobner bases for ideals in polynomial rings over a ring R, most of the theory and applications is developed for R — a field or a Noetherian commutative domain (e.g., for R a PID or a Dedekind domain); see [1], [2]. A von Neumann regular ring is another kind of zero-dimensional ring and here we prove that it is 1-Grobner, if it is commutative. Grobner basis for every finitely generated ideal in R[X], for a commutative von Neumann regular ring R. We recall several basic results concerning von Neumann regular rings and proceed to prove a couple of lemmas which allow us to prove Theorem 8 as follows: In a finitely generated ideal I of R[X], there is a polynomial whose leading coefficient divides all coefficients of any a polynomial of I. We give the basic definitions concerning the Grobner bases and present the important Lemma 9 This lemma enables us to prove the main result of the paper: every finitely generated ideal in R[X] has a strong Grobner basis. 3, and present all the steps for finding its Grobner basis

VON NEUMANN REGULAR RINGS

GRO BNER BASES