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Absolutely S-domains and pseudo-polynomial rings. (English) Zbl 1062.13500

From the paper: A domain \(R\) is said to be an S-domain if for each height 1 prime ideal \(p\) of \(R\), we have \(\text{ht}(p[X])=1\); \(R\) is called an absolutely S-domain (for short, AS-domain) if each domain \(T\) such that \(R\subseteq T\subseteq \text{qf}(R)\) 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 \(R/(q\cap R)\subseteq V/q\) is algebraic. A Noetherian domain \(R\) is an AS-domain if and only if \(\dim(R)\leq 1\). 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.

MSC:

13B02 Extension theory of commutative rings
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13G05 Integral domains
13A18 Valuations and their generalizations for commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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