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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Divisibility theory of semi-hereditary rings


Authors: P. N. Ánh and M. Siddoway
Journal: Proc. Amer. Math. Soc. 138 (2010), 4231-4242
MSC (2000): Primary 13A05, 13D05, 13F05; Secondary 06F05
Published electronically: July 9, 2010
MathSciNet review: 2680049
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Abstract: The semigroup of finitely generated ideals partially ordered by inverse inclusion, i.e., the divisibility theory of semi-hereditary rings, is precisely described by semi-hereditary Bezout semigroups. A Bezout semigroup is a commutative monoid $ S$ with 0 such that the divisibility relation $ a\vert b \Longleftrightarrow b\in aS$ is a partial order inducing a distributive lattice on $ S$ with multiplication distributive on both meets and joins, and for any $ a, b, d=a\wedge b\in S, a=da_1$ there is $ b_1\in S$ with $ a_1\wedge b_1=1, b=db_1$. $ S$ is semi-hereditary if for each $ a\in S$ there is $ e^2=e\in S$ with $ eS=a^{\perp}=\{x\in S \vert ax=0\}$. The dictionary is therefore complete: abelian lattice-ordered groups and semi-hereditary Bezout semigroups describe divisibility of Prüfer (i.e., semi-hereditary) domains and semi-hereditary rings, respectively. The construction of a semi-hereditary Bezout ring with a pre-described semi-hereditary Bezout semigroup is inspired by Stone's representation of Boolean algebras as rings of continuous functions and by Gelfand's and Naimark's analogous representation of commutative $ C^*$-algebras.


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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

M. Siddoway
Affiliation: Department of Mathematics and Computer Science, Colorado College, Colorado Springs, Colorado 80903
Email: msiddoway@coloradocollege.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10465-3
PII: S 0002-9939(2010)10465-3
Keywords: Bezout rings, Stone space, semi-hereditary, rings of continuous functions.
Received by editor(s): December 11, 2009
Received by editor(s) in revised form: February 25, 2010
Published electronically: July 9, 2010
Additional Notes: The first author was partially supported by the Hungarian National Foundation for Scientific Research grants no. K61007 and NK72523, Colorado College and UC-Colorado Springs during his stay at Colorado College in the Fall of 2006.
The second author was supported as the Verner Z. Reed Professor of Natural Science at Colorado College from 2007 to the present.
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.