Divisibility theory of semi-hereditary rings
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- by P. N. Ánh and M. Siddoway PDF
- Proc. Amer. Math. Soc. 138 (2010), 4231-4242 Request permission
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.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
- M. Siddoway
- Affiliation: Department of Mathematics and Computer Science, Colorado College, Colorado Springs, Colorado 80903
- Email: msiddoway@coloradocollege.edu
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 4231-4242
- MSC (2000): Primary 13A05, 13D05, 13F05; Secondary 06F05
- DOI: https://doi.org/10.1090/S0002-9939-2010-10465-3
- MathSciNet review: 2680049