The domain and prime properties for Koszul rings and algebras

TL;DR

This paper introduces a new technique to determine when Koszul graded rings are prime or domains by examining their Koszul duals, with applications to specific classes of Calabi-Yau algebras and preprojective algebras.

Contribution

It develops a categorical method to establish primeness and domain properties of Koszul rings using duality, extending previous approaches and applying to Calabi-Yau and preprojective algebras.

Findings

Koszul graded rings can be shown to be prime or domains via their duals.

Connected quivers with minimum degree ≥ 2 yield prime preprojective algebras.

The method applies to Koszul twisted Calabi-Yau algebras of dimension 2.

Abstract

We establish a technique to prove that a Koszul graded ring is prime or a domain using information about its Koszul dual. This is based on a general categorical result that expands on methods of J.Y. Guo, which proves that certain orbital rings are prime or domains. We apply this method to prove that if $A = kQ/I$ is a Koszul twisted Calabi-Yau algebra of dimension 2, such that $Q$ is connected with every vertex having outdegree at least 2, then $A$ is a prime piecewise domain. In particular, the preprojective algebra of a connected quiver whose underlying graph has minimum degree at least 2 is a prime piecewise domain.