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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.
Noomen Jarboui, and Ihsen Yengui. "Absolutely S-domains and pseudo-polynomial rings." Colloquium Mathematicae 94.1 (2002): 1-19. <http://eudml.org/doc/285065>.
@article{NoomenJarboui2002, 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.}, author = {Noomen Jarboui, Ihsen Yengui}, journal = {Colloquium Mathematicae}, keywords = {polynomial ring; Jaffard ring; survival extension; lying-over extension; absolutely S-domain; survival pair; lying-over pair}, language = {eng}, number = {1}, pages = {1-19}, title = {Absolutely S-domains and pseudo-polynomial rings}, url = {http://eudml.org/doc/285065}, volume = {94}, year = {2002}, }
TY - JOUR AU - Noomen Jarboui AU - Ihsen Yengui TI - Absolutely S-domains and pseudo-polynomial rings JO - Colloquium Mathematicae PY - 2002 VL - 94 IS - 1 SP - 1 EP - 19 AB - 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. LA - eng KW - polynomial ring; Jaffard ring; survival extension; lying-over extension; absolutely S-domain; survival pair; lying-over pair UR - http://eudml.org/doc/285065 ER -