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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 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 *semi-hereditary* if for each there is with . 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 -algebras.

**1.**D. D. Anderson,*Multiplication ideals, multiplication rings, and the ring 𝑅(𝑋)*, Canad. J. Math.**28**(1976), no. 4, 760–768. MR**0424794****2.**P. N. Ánh, L. Márki, P. Vámos, Divisibility theory in commutative rings: Bezout semigroups, manuscript.**3.**Bruno Bosbach,*Representable divisibility semigroups*, Proc. Edinburgh Math. Soc. (2)**34**(1991), no. 1, 45–64. MR**1093175**, 10.1017/S0013091500004995**4.**Ronald G. Douglas,*Banach algebra techniques in operator theory*, 2nd ed., Graduate Texts in Mathematics, vol. 179, Springer-Verlag, New York, 1998. MR**1634900****5.**Henri Cartan and Samuel Eilenberg,*Homological algebra*, Princeton University Press, Princeton, N. J., 1956. MR**0077480****6.**A. M. W. Glass and W. Charles Holland (eds.),*Lattice-ordered groups*, Mathematics and its Applications, vol. 48, Kluwer Academic Publishers Group, Dordrecht, 1989. Advances and techniques. MR**1036072****7.**Franz Halter-Koch,*A characterization of Krull rings with zero divisors*, Arch. Math. (Brno)**29**(1993), no. 1-2, 119–122. MR**1242634****8.**Franz Halter-Koch,*Ideal systems*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 211, Marcel Dekker, Inc., New York, 1998. An introduction to multiplicative ideal theory. MR**1828371****9.**Franz Halter-Koch,*Construction of ideal systems with nice Noetherian properties*, Non-Noetherian commutative ring theory, Math. Appl., vol. 520, Kluwer Acad. Publ., Dordrecht, 2000, pp. 271–285. MR**1858166****10.**M. Henriksen and M. Jerison,*The space of minimal prime ideals of a commutative ring*, Trans. Amer. Math. Soc.**115**(1965), 110–130. MR**0194880**, 10.1090/S0002-9947-1965-0194880-9**11.**Nathan Jacobson,*Structure of rings*, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR**0222106****12.**Paul Jaffard,*Les systèmes d’idéaux*, Travaux et Recherches Mathématiques, IV, Dunod, Paris, 1960 (French). MR**0114810****13.**Chr. U. Jensen,*A remark on arithmetical rings*, Proc. Amer. Math. Soc.**15**(1964), 951–954. MR**0179197**, 10.1090/S0002-9939-1964-0179197-5**14.**Eben Matlis,*The minimal prime spectrum of a reduced ring*, Illinois J. Math.**27**(1983), no. 3, 353–391. MR**698302****15.**M. H. Stone,*Applications of the theory of Boolean rings to general topology*, Trans. Amer. Math. Soc.**41**(1937), no. 3, 375–481. MR**1501905**, 10.1090/S0002-9947-1937-1501905-7

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
13A05,
13D05,
13F05,
06F05

Retrieve articles in all journals with MSC (2000): 13A05, 13D05, 13F05, 06F05

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

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.