Absolutely S-domains and pseudo-polynomial rings

Volume 94 / 2002

Noomen Jarboui, Ihsen Yengui Colloquium Mathematicum 94 (2002), 1-19 MSC: Primary 13B02; Secondary 13C15, 13A17, 13A18, 13B25, 13E05. DOI: 10.4064/cm94-1-1

Abstract

A domain $R$ is called an absolutely S-domain (for short, AS-domain) if each domain $T$ such that $R\subseteq T\subseteq \mathop {\rm qf}(R)$ 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 $\mathop {\rm dim}\nolimits (R)\leq 1$. 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.

Authors

  • Noomen JarbouiDepartment of Mathematics
    Faculty of Sciences
    University of Sfax
    B.P. 802, 3018 Sfax, Tunisia
    e-mail
  • Ihsen YenguiDepartment of Mathematics
    Faculty of Sciences B.P. 802, 3018 Sfax, Tunisia
    University of Sfax
    e-mail

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