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On pseudo-almost valuation domains. (English) Zbl 1113.13001

Summary: Let \(R\) be an integral domain with quotient field \(K\) and integral closure \(R'\). D. D. Anderson and M. Zafrullah [J. Algebra 142, No. 2, 285–309 (1991; Zbl 0749.13013)] called \(R\) an “almost valuation domain” if for every nonzero \(x\in K\), there is a positive integer \(n\) such that either \(x^n \in R\) or \(x^{-n} \in R\). In this article, we introduce a new closely related class of integral domains. We define a prime ideal \(P\) of \(R\) to be a “pseudo-strongly prime ideal” if, whenever \(x, y \in K\) and \(xyP \subseteq P\), then there is a positive integer \(m \geq 1\) such that either \(x^m\in R\) or \(y^m P \subseteq P\). If each prime ideal of \(R\) is a pseudo-strongly prime ideal, then \(R\) is called a “pseudo-almost valuation domain” (PAVD).
We show that the class of valuation domains, the class of pseudo-valuation domains, the class of almost valuation domains, and the class of almost pseudo-valuation domains are properly contained in the class of pseudo-almost valuation domains; also we show that the class of pseudo-almost valuation domains is properly contained in the class of quasilocal domains with linearly ordered prime ideals. Among the properties of PAVDs, we show that an integral domain \(R\) is a PAVD if and only if for every nonzero \(x\in K\), there is a positive integer \(n\geq 1\) such that either \(x^n \in R\) or \(ax^{-n} \in R\) for every nonunit \(a\in R\). We show that pseudo-almost valuation domains are precisely the pullbacks of almost valuation domains, we characterize pseudo-almost valuation domains of the form \(D+M\), and we use this characterization to construct PAVDs that are not almost valuation domains. We show that if \(R\) is a noetherian PAVD, then \(R\) has Krull dimension at most one and \(R'\) is a valuation domain; we show that every overring of a PAVD \(R\) is a PAVD iff \(R'\) is a valuation domain and every integral overring of \(R\) is a PAVD.
For Part II, see [G. W. Chang, Korean J. Math. 19, No. 4, 343–349 (2011; Zbl 1516.13002)].

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13A10 Radical theory on commutative rings (MSC2000)
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
14G05 Rational points
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[1] DOI: 10.1016/0021-8693(91)90309-V · Zbl 0749.13013 · doi:10.1016/0021-8693(91)90309-V
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[4] DOI: 10.4153/CJM-1980-029-2 · Zbl 0406.13001 · doi:10.4153/CJM-1980-029-2
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[9] Dobbs D. E., Houston J. Math. 4 pp 551– (1978)
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