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Keywords:
valuation monoids; Prüfer domains
valuation monoids; Prüfer domains
Summary:
It is well known that an integral domain is a valuation domain if and only if it possesses only one finitary ideal system (Lorenzen $r$-system of finite character). We prove an analogous result for root-closed (cancellative) monoids and apply it to give several new characterizations of Prüfer (multiplication) monoids and integral domains.
References:
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[5] Halter-Koch F.: Characterization of Prüfer multiplication monoids and domains by means of spectral module systems. Monatsh. Math. 139 (2003), 19–31. MR 1981115 | Zbl 1058.20049
[6] Halter-Koch F.: Valuation Monoids, Defining Systems and Approximation Theorems. Semigroup Forum 55 (1997), 33–56. MR 1446657 | Zbl 0880.20047
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