Divisibility theory in commutative rings: Bezout monoids
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- by P. N. Ánh, L. Márki and P. Vámos PDF
- Trans. Amer. Math. Soc. 364 (2012), 3967-3992 Request permission
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 $d=x\wedge y$ and $dx_1=x$, then there is a $y_1\in S$ with $dy_1=y$ and $x_1\wedge y_1=1$. 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.References
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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
- 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 - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2912441
Dedicated: Dedicated to Bruno Bosbach