Good rings and homogeneous polynomials
Language
en
Document de travail - Pré-publication
English Abstract
In 2011, Khurana, Lam and Wang define 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 are an integer N = N (a, b) ≥ 1 and ...Read more >
In 2011, Khurana, Lam and Wang define 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 are an integer N = N (a, b) ≥ 1 and λ = λ(a, b) ∈ A such that b N + λa ∈ A × , the unit group of A. In 2019, Berman and Erman consider rings with the following property (**) A commutative unital ring A has enough homogeneous polynomials if for any k ≥ 1 and set S := {p 1 , p 2 , ..., p k } , of primitive points in A n and any n ≥ 2, there exists an homogeneous polynomial P (X 1 , X 2 , ..., X n) ∈ A[X 1 , X 2 , ..., X n ]) with deg P ≥ 1 and P (p i) ∈ A × for 1 ≤ i ≤ k. We show in this article 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 we built by Goldman in 1963,we show that the converse is false.Read less <
Origin
Hal imported