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 $a,b\in A$ with $aA+bA=A$ there is an integer $N=N(a,b)\ge 1$ and $\lambda =\lambda (a,b)\in A$ such that $b^N+\lambda a\in A^{\times }$, 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 $k\ge 1$ and set $S:=\{p_1,p_2,\dots ,p_k\}$, of primitive points in $A^n$ and any $n\ge 2$, there exists a homogeneous polynomial $P(X_1,X_2,\dots ,X_n)\in A[X_1,X_2,\dots ,X_n]$ with $\text{deg }P\ge 1$ and $P(p_i)\in A^{\times }$ for $1\le i\le k$.

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.