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Divisibility theory in commutative rings: Bezout monoids


Authors: P. N. Ánh, L. Márki and P. Vámos
Journal: Trans. Amer. Math. Soc. 364 (2012), 3967-3992
MSC (2010): Primary 06F05; Secondary 13A05, 13F05, 20M14
DOI: https://doi.org/10.1090/S0002-9947-2012-05424-9
Published electronically: March 22, 2012
MathSciNet review: 2912441
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Abstract: A ubiquitous class of lattice ordered semigroups introduced by Bosbach in 1991, which we will call Bezout monoids, seems to be the appropriate structure for the study of divisibility in various classical rings like GCD domains (including UFD's), rings of low dimension (including semihereditary rings), as well as certain subdirect products of such rings and certain factors of such subdirect products. A Bezout monoid is a commutative monoid $ S$ with 0 such that under the natural partial order (for $ a,b\in S$, $ a\leq b\in S\Longleftrightarrow \,\, bS\subseteq aS$), $ S$ is a distributive lattice, multiplication is distributive over both meets and joins, and for any $ x,\, y\in S$, if $ d=x\mathop {\wedge } y$ and $ dx_1=x$, then there is a $ y_1\in S$ with $ dy_1=y$ and $ x_1\mathop {\wedge } y_1=1$. In the present paper, Bezout monoids are investigated by using filters and $ m$-prime filters. We also prove analogues of the Pierce and the Grothendieck sheaf representations of rings for Bezout monoids. The question as to whether Bezout monoids describe divisibility in Bezout rings (rings whose finitely generated ideals are principal) is still open.


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Additional Information

P. N. Ánh
Affiliation: Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1364 Budapest, Pf. 127 Hungary
Email: anh@renyi.hu

L. Márki
Affiliation: Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1364 Budapest, Pf. 127 Hungary
Email: marki@renyi.hu

P. Vámos
Affiliation: Department of Mathematics, University of Exeter, North Park Road, Exeter, EX4 4QF, England
Email: p.vamos@exeter.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2012-05424-9
Keywords: Bezout ring, divisibility semigroup, Bezout monoid, spectrum, prime filter, sheaf.
Received by editor(s): June 3, 2010
Published electronically: March 22, 2012
Additional Notes: The first author was partially supported by the Hungarian National Foundation for Scientific Research grant no. K61007 and by the Colorado College and the University of Colorado at Colorado Springs during his stay at Colorado College in the fall of 2006
The second author was partially supported by the Hungarian National Foundation for Scientific Research grant no. NK72523
The third author acknowledges the hospitality of the Rényi Institute during the initial research phase of this paper
Dedicated: Dedicated to Bruno Bosbach
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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