# Good rings and homogeneous polynomials

## TL;DR

This paper proves the equivalence of two properties related to rings, introduces the concept of good rings, and explores their characteristics through examples including rings of pictorsion and Dedekind domains.

## Contribution

It establishes the equivalence of properties (*) and (**) defining good rings and analyzes their presence in rings of pictorsion and Dedekind domains.

## Key findings

- Properties (*) and (**) are equivalent for commutative unital rings.
- Rings of pictorsion are examples of good rings.
- Dedekind domains can serve as counterexamples to the converse.

## 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 $\in$ A with aA + bA = A there are an integer N = N (a, b) $\ge$ 1 and $\lambda$ = $\lambda$(a, b) $\in$ A such that b N + $\lambda$a $\in$ A x , 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 $\ge$ 1 and set S := {p 1 , p 2 , ..., p k } , of primitive points in A n and any n $\ge$ 2, there exists an homogeneous polynomial P (X 1 , X 2 , ..., X n) $\in$ A[X 1 , X 2 , ..., X n ]) with deg P $\ge$ 1 and P (p i) $\in$ A x for 1 $\le$ i $\le$ 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.