Pullback diagrams and Kronecker function rings (Q2288242)

Let \(R \subseteq S\) be an extension of commutative rings with identity, \(J(R)\) be the Jacobson radical of \(R\), \(\mathfrak{I}(R,S)\) be the set of \(R\)-submodules of \(S\), \(X\) be an indeterminate over \(S\), and \(S[X]\) be the polynomial ring over \(S\). For each \(f \in S[X]\), let \(c_A(f)\) be the \(A\)-submodule of \(S\) generated by the coefficients of \(f\). Then \(S(X) = \{f/g \mid f,g \in S[X]\) and \(c_S(g)=S\}\) is an overring of \(S[X]\). Let \(\star\) be a star operation on \(R \subseteq S\) as in [M. Knebush and T. Kaiser, Manis valuations and Prüfer extensions, II, Lectures notes in mathematics 2103, Springer, Cham, Switzerland, 2014]. Then \(Kr(\star) = \{f/g \mid f,g \in S[X], c_S(g)=S\) and \((c_R(f)H)^{\star} \subseteq (c_R(g)H)^{\star}\) for some finitely generated \(R\)-submodule \(H\) of \(S\) with \(HS= S\}\) is called a Kronecker subring of \(S(X)\) over \(R\). Furthermore, let \(I\) be a common ideal of \(R\) and \(S\) such that \(I\) is a maximal ideal of \(S\), and consider the following pullback diagram of type \((\Box)\); \(R/I \twoheadleftarrow R \hookrightarrow S \twoheadrightarrow^{\psi} S/I \hookleftarrow R/I\), where \(S \twoheadrightarrow^{\psi} S/I\) denotes the natural epimorphism \(\psi: S \rightarrow S/I\) given by \(\psi(a) = a+I\). Let \(R \subseteq S\) be a Prüfer extension, so every finitely generated ideal \(I\) of \(R\) with \(IS=S\) is \(S\)-invertible, i.e., \(IJ = R\) for some \(R\)-submodule \(J\) of \(S\). In this paper the authors show that \(IKr(\star) = IS(X)\). Let \(\star'\) be a star operation on \(R/I \subseteq S/I\) such that \(Kr(\star') \leftarrow Kr(\star) \hookrightarrow S(X) \twoheadrightarrow^{\bar{\psi}} (S/I)(X) \hookleftarrow Kr(\star')\) commutes and \(\psi(A^{\star}) = \psi(A)^{\star'}\) for each \(A \in \mathfrak{I}(R,S)\). They also show that if \(I \subseteq J(R)\), then \(Kr(\star') \cong Kr(\star)/IKr(\star)\). Finally, they note that if \(\star_1\) is a star operation on \(R/I \subseteq S/I\) and if \(A^{\star_2} = \psi^{-1}(\psi(A)^{\star_1})\) for each \(A \in \mathfrak{I}(R,S)\), then \(\star_2\) is a star operation on \(R \subseteq S\) with \(\psi(A)^{\star_1} = \psi(A^{\star_2})\). Then they prove that if \(I \subseteq J(R)\), then the diagram \(Kr(\star_1) \twoheadleftarrow Kr(\star_2) \hookrightarrow S(X) \twoheadrightarrow^{\bar{\psi}} (S/I)(X) \hookleftarrow Kr(\star_1)\) is a pullback diagram of type \((\Box)\). In fact, the authors prove that the above results are true without the assumption that \(R \subseteq S\) is Prüfer and \(I \subseteq J(R)\). However, they use the properties of Prüfer extensions and \(I \subseteq J(R)\) in order to prove these results. More precisely, the Prüfer assumption is necessary for Proposition 2.2, Proposition 2.6, Theorem 2.13, and Corollary 2.14. The \(I \subseteq J(R)\) assumption is necessary for Proposition 2.6 and Theorem 2.12.