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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Infinitely generated projective modules over pullbacks of rings


Authors: Dolors Herbera and Pavel Příhoda
Journal: Trans. Amer. Math. Soc. 366 (2014), 1433-1454
MSC (2010): Primary 16L30, 16D40
Published electronically: August 16, 2013
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Abstract: We use pullbacks of rings to realize the submonoids $ M$ of $ (\mathbb{N} _0\cup \{\infty \})^k$, which are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated projective right $ R$-modules over a suitable semilocal ring. For these rings, the behavior of countably generated projective left $ R$-modules is determined by the monoid $ D(M)$ defined by reversing the inequalities determining the monoid $ M$. These two monoids are not isomorphic in general. As a consequence of our results we show that there are semilocal rings such that all its projective right modules are free but this fails for projective left modules. This answers in the negative a question posed by Fuller and Shutters. We also provide a rich variety of examples of semilocal rings having nonfinitely generated projective modules that are finitely generated modulo the Jacobson radical.


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Additional Information

Dolors Herbera
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Email: dolors@mat.uab.cat

Pavel Příhoda
Affiliation: Faculty of Mathematics and Physics, Department of Algebra, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic
Email: prihoda@karlin.mff.cuni.cz

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05798-4
PII: S 0002-9947(2013)05798-4
Received by editor(s): December 1, 2011
Received by editor(s) in revised form: January 16, 2012, and January 23, 2012
Published electronically: August 16, 2013
Additional Notes: The final version of this paper was written while the first author was visiting NTNU (Trondheim, Norway). She thanks her host for the kind hospitality. She was also partially supported by DGI MICIIN (Spain) through Project MTM2011-28992-C02-01, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya through Project 2005SGR00206
The second author was supported by GAČR 201/09/0816 and research project MSM 0021620839
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.