Real algebraic geometry for matrices over commutative rings

TL;DR

This paper extends fundamental theorems of real algebraic geometry to matrix rings over commutative rings, revealing new insights into preorderings and orderings that differ from classical cases.

Contribution

It introduces a novel framework for preorderings on matrix rings, extending key theorems like Artin-Lang and Krivine-Stengle to these non-commutative settings.

Findings

Orderings of M_n(R) correspond to those of R.

Preorderings of M_n(R) are not in one-to-one correspondence with those of R.

The theory is not Morita equivalent to classical real algebraic geometry.

Abstract

We define and study preorderings and orderings on rings of the form $M_n(R)$ where $R$ is a commutative unital ring. We extend the Artin-Lang theorem and Krivine-Stengle Stellens\"atze (both abstract and geometric) from $R$ to $M_n(R)$. While the orderings of $M_n(R)$ are in one-to-one correspondence with the orderings of $R$, this is not true for preorderings. Therefore, our theory is not Morita equivalent to the classical real algebraic geometry.