Some remarks on the ring \(R^\#\) (Q2785911)

Let \(R\) be an integral domain with quotient field \(K\). In this paper, we continue the study of a certain overring of \(R\), \(R^\# =\bigcap \{R_x\mid x\) is a nonzero nonunit of \(R\}\), begun by \textit{D. F. Anderson} and \textit{A. Bouvier} in Ann. Univ. Ferrara, Nuova Ser., Sez. VII 32, 15-38 (1986; Zbl 0655.13002) (here, \(R_x= R[1/x])\). If \(R\) is not quasilocal, then \(R^\# =R\); and \(R^\# =K\) when \(R\) is a one-dimensional quasilocal integral domain [loc. cit.; theorem 1.2]. Hence we are interested in the case when \(R\) is a quasilocal integral domain with maximal ideal \(M\) and \(\dim R\geq 2\). In the quasilocal case, \(R^\# =\bigcap \{Rp \mid P\in \text{spec} (R)- \{M\}\}\). Also, \(R^\# =S(M)\), where \(S(M)\) is the \(S\)-transform of \textit{J. H. Hays} [J. Algebra 57, 223-229 (1979; Zbl 0427.13007)]; and if \(M\) is finitely generated, then \(R^\# =T(M)\), the usual ideal transform of \textit{M. Nagata} [Mem. Coll. Sci., Univ. Kyoto, Ser. A 30, 57-70 (1956; Zbl 0089.02501)]. Finally, if \(R\) is a Krull domain which is not a DVR, then \(R^\# =R\).NEWLINENEWLINENEWLINEIn the first section, we determine \(R^\#\) when \(R\) is a pullback. The second section relates \(\text{spec} (R^\# [X_1, \dots, X_n])\) to \(\text{spec} (R[X_1, \dots, X_n])\). The third section investigates \(R^\#\) when \(R\) is a Jaffard domain, and the fourth section gives necessary and sufficient conditions for \(R^\#\) to be a proper localization of \(R\). In the final section, we study Cl\((R^\#)\), the \((t\)-)class group of \(R^\#\).NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].