Absolutely S-domains and pseudo-polynomial rings

Abstract

Abstract.A domain R is called an absolutely S-domain (for short, AS-domain) if each domain T such that R ⊆ T ⊆ qf(R) is an S-domain. We show that R is an AS-domain if and only if for each valuation overring V of R and each height one prime ideal q of V , the extension R/(q ∩ R) ⊆ V /q is algebraic. A Noetherian domain R is an AS-domain if and only if dim(R) ≤ 1. In Section 2, we study a class of R-subalgebras of R [X] which share many spectral properties with the polynomial ring R[X] and which we call pseudo-polynomial rings. Section 3 is devoted to an affirmative answer to D. E. Dobbs's question of whether a survival pair must be a lying-over pair in the case of transcendental extension. Introduction.In this paper, all rings considered are commutative with identity. An inclusion of rings signifies that the smaller ring is a subring of the larger and has the same identity. Let R be a ring and n a positive integer. We denote by R[n] the ring of polynomials in n indeterminates over R and by R[X] the ring of polynomials in one indeterminate. We denote by dim(R) the Krull dimension of R and by dim v (R) its valuative dimension, that is, the limit of the sequence (dim(R[n]) − n, n ∈ N). If p is a prime ideal of R, we denote by ht(p) the height of p, and by ht v (p) the limit of the sequence (ht(p[n]), n ∈ N).Given a finite-dimensional ring R, we say that R is a Jaffard ring if dim(R) = dim v (R) [2]. This property is not local; we say that R is a locally Jaffard ring if R p is a Jaffard ring for each prime ideal p of R. A domain R is said to be an S-domain if for each height 1 prime ideal p of R, we have ht(p[X]) = 1. A strong S-ring is a ring R such that for each prime ideal p of R, R/p is an S-domain; equivalently for any consecutive primes p ⊂ q in R,. An overring of a domain R is a ring contained between R and its quotient field qf(R).For an extension of domains R ⊆ T , we denote by tr.deg[T : R] the transcendence degree of qf(T ) over qf(R). Recall that an extension R ⊆ T 2000 Mathematics Subject Classification: Primary 13B02; Secondary 13C15, 13A17, 13A18, 13B25, 13E05.

When is each proper overring of $R$ an S(Eidenberg)-domain?

“…They characterized these domains in terms of pseudo-valuation domains. On the other hand the author and I. Yengui in [11] studied the domains R such that each domain contained between R and its quotient field is an S-domain. They are said to be absolutely S-domains.…”