Divisibility theory of semihereditary rings
Authors:
P. N. Ánh and M. Siddoway
Journal:
Proc. Amer. Math. Soc. 138 (2010), 42314242
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 semihereditary rings, is precisely described by semihereditary Bezout semigroups. A Bezout semigroup is a commutative monoid with 0 such that the divisibility relation is a partial order inducing a distributive lattice on with multiplication distributive on both meets and joins, and for any there is with . is semihereditary if for each there is with . The dictionary is therefore complete: abelian latticeordered groups and semihereditary Bezout semigroups describe divisibility of Prüfer (i.e., semihereditary) domains and semihereditary rings, respectively. The construction of a semihereditary Bezout ring with a predescribed semihereditary 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 algebras.
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 H. Cartan and S. Eilenberg, Homological algebra, Princeton, 1956. MR 0077480 (17:1040e)
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 A. M. W. Glass and W. Charles Holland (eds), Latticeordered groups: advances and techniques, Kluwer Academic Publishers, 1989. MR 1036072 (91i:06017)
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 M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375481. MR 1501905
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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/S000299392010104653
PII:
S 00029939(2010)104653
Keywords:
Bezout rings,
Stone space,
semihereditary,
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 UCColorado 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 HuisgenZimmermann
Article copyright:
© Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
