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Pure Projective Approximations

Published online by Cambridge University Press:  01 January 2009

IVO HERZOG  and
PHILIPP ROTHMALER
IVO HERZOG
Affiliation:
The Ohio State University at Lima, Lima, OH 45804, U.S.A. e-mail: [email protected]
PHILIPP ROTHMALER
Affiliation:
The City University of New York, University Ave and W 181 St, Bronx, NY 10453, U.S.A. e-mail: [email protected]
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Abstract

A notion of good behavior is introduced for a definable subcategory of left R-modules. It is proved that every finitely presented left R-module has a pure projective left -approximation if and only if the associated torsion class of finite type in the functor category (mod-R, Ab) is coherent, i.e., the torsion subobject of every finitely presented object is finitely presented. This yields a bijective correspondence between such well-behaved definable subcategories and preenveloping subcategories of the category Add(R-mod) of pure projective left R-modules. An example is given of a preenveloping subcategory ⊆ Add(R-mod) that does not arise from a covariantly finite subcategory of finitely presented left R-modules. As a general example of this good behavior, it is shown that if R is a ring over which every left cotorsion R-module is pure injective, then the smallest definable subcategory (R-proj) containing every finitely generated projective module is well-behaved.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 1 , January 2009 , pp. 83 - 94
Copyright
Copyright © Cambridge Philosophical Society 2008

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