Winter 2022 Good rings and homogeneous polynomials
Jean Fresnel, Michel Matignon
J. Commut. Algebra 14(4): 527-552 (Winter 2022). DOI: 10.1216/jca.2022.14.527


In 2011, Khurana, Lam and Wang defined the following property:

(*) A commutative unital ring A satisfies the property “power stable range one” if for all a,bA with aA+bA=A there is an integer N=N(a,b)1 and λ=λ(a,b)A such that bN+λaA×, the unit group of A.

In 2019, Berman and Erman considered rings with the following property:

(**) A commutative unital ring A has enough homogeneous polynomials if for any k1 and set S:={p1,p2,,pk}, of primitive points in An and any n2, there exists a homogeneous polynomial P(X1,X2,,Xn)A[X1,X2,,Xn] with deg P1 and P(pi)A× for 1ik.

We show that the two properties (*) and (**) are equivalent and we shall call a commutative unital ring with these properties a good ring.

When A is a commutative unital ring of pictorsion as defined by Gabber, Lorenzini and Liu in 2015, we show that A is a good ring. Using a Dedekind domain built by Goldman in 1963, we show that the converse is false.


Download Citation

Jean Fresnel. Michel Matignon. "Good rings and homogeneous polynomials." J. Commut. Algebra 14 (4) 527 - 552, Winter 2022.


Received: 13 December 2019; Revised: 29 December 2020; Accepted: 10 January 2021; Published: Winter 2022
First available in Project Euclid: 15 November 2022

MathSciNet: MR4509406
zbMATH: 1510.13002
Digital Object Identifier: 10.1216/jca.2022.14.527

Primary: 13A99
Secondary: 14G99

Keywords: enough homogeneous polynomials , good point , good ring , pictorsion ring , primitive point

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium


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Vol.14 • No. 4 • Winter 2022
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