Previous |  Up |  Next


valuation monoids; Prüfer domains
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.
[1] Aubert K. E.: Some characterizations of valuation rings. Duke Math. J. 21 (1954), 517–525. MR 0062727
[2] Garcia J. M., Jaros P., Santos E.: Prüfer $*$-multiplication domains and torsion theories. Comm. Algebra 27 (1999), 1275–1295. MR 1669156
[3] Halter-Koch F.: Ideal Systems. Marcel Dekker 1998. MR 1828371 | Zbl 0953.13001
[4] Halter-Koch F.,: Construction of ideal systems having nice noetherian properties. Commutative Rings in a Non-Noetherian Setting (S. T. Chapman and S. Glaz, eds.), Kluwer 2000, 271–285. MR 1858166
[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
Partner of
EuDML logo