Finitely generated intermediate rings
Let R be a commutative unitary ring and let S be an affine extension of R. If each intermediate ring T, \(R\subseteq T\subseteq S\), is affine over R, then S is said to be strongly affine over R. This condition implies that tr.deg.S/R\(\leq 1\), and if equality holds, then R is Noetherian and von Neumann regular. This paper investigates various questions concerning strongly affine extensions. Primary emphasis is on the case where R is Noetherian, in which case (R,S) is a Noetherian pair in the sense of \textit{A. R. Wadsworth} [Trans. Am. Math. Soc. 195, 201-211 (1974; Zbl 0294.13010)]. While the converse fails in general, it is shown that if R is a Noetherian universally Japanese ring, if S/R is affine, and if (R,S) is a Noetherian pair, then S is strongly affine over R. One section of the paper is devoted to a consideration of the strongly affine condition in the case where R is an integral domain and S is an overring of R, particularly for the case where R is a Prüfer domain. For example, if R is a Prüfer domain satisfying a.c.c. on prime ideals, then each affine overring of R is strongly affine over R if and only if each nonzero ideal of R is contained in only finitely many prime ideals of R. Conversely, an integrally closed domain R is a Prüfer domain if each affine overring is strongly affine over R. However, the following question remains open: If each affine overring of the domain D is strongly affine over D, is the integral closure of D a Prüfer domain?
scientific article; zbMATH DE number 3899041
scientific article