The center of the total ring of fractions

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 $\mathrm{GKdim}(A)<\mathrm{GKdim}(Z(A))+1$, 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.