J. Commut. Algebra 14 (4), 527-552, (Winter 2022) DOI: 10.1216/jca.2022.14.527
Jean Fresnel, Michel Matignon
KEYWORDS: primitive point, good point, good ring, enough homogeneous polynomials, pictorsion ring, 13A99, 14G99
In 2011, Khurana, Lam and Wang defined the following property:
(*) A commutative unital ring satisfies the property “power stable range one” if for all with there is an integer and such that , the unit group of .
In 2019, Berman and Erman considered rings with the following property:
(**) A commutative unital ring has enough homogeneous polynomials if for any and set , of primitive points in and any , there exists a homogeneous polynomial with and for .
We show that the two properties (*) and (**) are equivalent and we shall call a commutative unital ring with these properties a good ring.
When is a commutative unital ring of pictorsion as defined by Gabber, Lorenzini and Liu in 2015, we show that is a good ring. Using a Dedekind domain built by Goldman in 1963, we show that the converse is false.