Good rings and homogeneous polynomials

Abstract

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,b∈A with aA+bA=A there is an integer N=N(a,b)≥1 and λ=λ(a,b)∈A such that bN+λa∈A×, 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 k≥1 and set S:={p1,p2,…⁡,pk}, of primitive points in An and any n≥2, there exists a homogeneous polynomial P(X1,X2,…⁡,Xn)∈A[X1,X2,…⁡,Xn] with deg P≥1 and P(pi)∈A× for 1≤i≤k. 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.