Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids

Pere Ara; Alberto Facchini

doi:10.1515/FORUM.2006.021

Published by De Gruyter May 18, 2006

Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids

Pere Ara and Alberto Facchini

From the journal Forum Mathematicum

https://doi.org/10.1515/FORUM.2006.021

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Abstract

We show that a number of pullback diagrams appear naturally in the study of preordered Grothendieck groups. The passage of projective modules from a ring R to a factor ring R/I turns out to be particularly good for a certain class of ideals, which we call almost trace ideals. We generalize to arbitrary rings a result by Goodearl concerning the lattice of the directed convex subgroups of K₀(R). Finally, we show that a variant (I) of the Grothendieck group of I, introduced by Quillen, has an easy description in terms of projective modules when I is an almost trace ideal.

(Communicated by Rüdiger Göbel)

* Partially supported by the DGI and the European Regional Development Fund, jointly, through Project BFM2002-01390, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.

† Partially supported by Ministero dell'Istruzione, dell'Università e della Ricerca (Italy) and by Departament d'Universitats, Recerca i Societat de la Informació (Generalitat de Catalunya). This paper was written during a sabbatical year of the second author at the Centre de Recerca Matemàtica (Barcelona). He acknowledges the kind hospitality received.

Received: 2004-02-02

Revised: 2004-07-14

Published Online: 2006-05-18

Published in Print: 2006-05-01

Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids

Abstract

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