The center of the total ring of fractions

TL;DR

This paper proves that under certain conditions, the center of the total ring of fractions of a right Ore domain is isomorphic to the field of fractions of its center, with examples mainly from skew PBW extensions.

Contribution

It establishes a new isomorphism between the center of the total ring of fractions and the field of fractions of the center for specific algebras.

Findings

Center of total ring of fractions is isomorphic to the field of fractions of the center.

The result applies to finitely generated $K$-algebras with GK dimension constraints.

Includes numerous examples, especially within skew PBW extensions.

Abstract

Let $A$ be a right Ore domain, $Z(A)$ be the center of $A$ and $Q_r(A)$ be the right total ring of fractions of $A$. If $K$ is a field and $A$ is a $K$-algebra, in this short paper we prove that if $A$ is finitely generated and ${\rm GKdim}(A)<{\rm GKdim}(Z(A))+1$, then $Z(Q_r(A))\cong Q(Z(A))$. Many examples that illustrate the theorem are included, most of them within the skew $PBW$ extensions.