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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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.
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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

  • Dedicated: Dedicated to Bruno Bosbach
  • © 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