Abstract
We prove that the multiplicative monoid of principal ideals partially ordered by reverse inclusion, called the divisibility theory, of a Bezout ring R with one minimal prime ideal is a factor of the positive cone of a lattice-ordered abelian group by an appropriate filter if the localization of R at its minimal prime ideal is not a field. This result extends a classical result of Clifford (Am. J. Math. 76:631–646, 1954) saying that the divisibility theory of a valuation ring is a Rees factor of the positive cone of a totally ordered abelian group and suggests to modify Kaplansky’s (later disproved) conjecture (Fuchs and Salce, Mathematical Surveys and Monographs 84, 2001) as to whether a Bezout ring whose localization at every minimal prime ideal has at least three ideals is the factor of an appropriate Bezout domain.
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Acknowledgments
The author was partially supported by both the Hungarian National Foundation for Scientific Research grants no. K-101515, Institute of Mathematics (Viện Toán Học 18 Hoàng Quôc Việt, Hà Nội) and VIASM (Vietnamese Institute of Advanced Study in Mathematics) for his stay in Hanoi, Vietnam.
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Ánh, P.N. Generalization of a Theorem of Clifford. Acta Math Vietnam 41, 471–480 (2016). https://doi.org/10.1007/s40306-015-0159-3
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DOI: https://doi.org/10.1007/s40306-015-0159-3