Pullback diagrams and Kronecker function rings

Abstract

Knebusch and Kaiser introduced the notion of Kronecker function ring of a ring extension with respect to a star operation to generalize the classical notion of Kronecker function ring. Let $\star $ be a star operation on the extension $R\subseteq S$. Let $Kr ( \star)$ be the set of all quotients ${f}/{g}\in S(X)$ such that $f, g\in S[X]$, $c_{S}(g) = S$ and $(c_{R}(f)H)^{\star } \subseteq (c_{R}(g)H)^{\star }$ for some finitely generated $S$-regular $R$-submodule $H$ of $S$, where for each ring $A$ such that $R\subseteq A \subseteq S$, $c_{A}(g)$ denotes the content of $g\in S[X]$. The ring $Kr (\star )$ is a Kronecker subring of $S(X)$ by a theorem of Knebusch and Kaiser. We study properties of the ring $Kr (\star )$ in pullback diagrams.