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ON KRULL–SCHMIDT FINITELY ACCESSIBLE CATEGORIES

Part of: Categories and theories

Published online by Cambridge University Press:  01 April 2011

SEPTIMIU CRIVEI
SEPTIMIU CRIVEI*
Affiliation:
Faculty of Mathematics and Computer Science, ‘Babeş-Bolyai’ University, Str. M. Kogălniceanu 1, 400084 Cluj-Napoca, Romania (email: crivei@math.ubbcluj.ro)
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Abstract

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Let 𝒞 be a finitely accessible additive category with products, and let (Ui)i∈I be a family of representative classes of finitely presented objects in 𝒞 such that each object Ui is pure-injective. We show that 𝒞 is a Krull–Schmidt category if and only if every pure epimorphic image of the objects Ui is pure-injective.

Keywords

Krull–Schmidt categorypure-injective objectsemiperfect ringOsofsky theorem

MSC classification

Secondary: 18C35: Accessible and locally presentable categories
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 1 , August 2011 , pp. 90 - 97
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Adámek, J. and Rosický, J., Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series, 189 (Cambridge University Press, Cambridge, 1994).CrossRefGoogle Scholar
[2]Ánh, P. N., Loi, N. V. and Thanh, D. V., ‘Perfect rings without identity’, Comm. Algebra 19 (1991), 1069–1082.CrossRefGoogle Scholar
[3]Bühler, T., ‘Exact categories’, Expo. Math. 28 (2010), 1–69.CrossRefGoogle Scholar
[4]Cárceles, A. I., ‘Inmersiones de categorías finitamente accesibles y categorías exactamente definibles’, Dualidad y simetría en categorías, University of Murcia, Spain, 2008 (in Spanish).Google Scholar
[5]Castaño Iglesias, F., Chifan, N. and Năstăsescu, C., ‘Localization and colocalization on certain Grothendieck categories’, Acta Math. Sin. 25 (2009), 379–392.CrossRefGoogle Scholar
[6]Crawley-Boevey, W., ‘Locally finitely presented additive categories’, Comm. Algebra 22 (1994), 1641–1674.CrossRefGoogle Scholar
[7]Crivei, S., Năstăsescu, C. and Torrecillas, B., ‘On the Osofsky–Smith theorem’, Glasg. Math. J. 52A (2010), 61–67.CrossRefGoogle Scholar
[8]Crivei, S., Prest, M. and Torrecillas, B., ‘Covers in finitely accessible categories’, Proc. Amer. Math. Soc. 138 (2010), 1213–1221.CrossRefGoogle Scholar
[9]Dung, N. V. and García, J. L., ‘Additive categories of locally finite representation type’, J. Algebra 238 (2001), 200–238.CrossRefGoogle Scholar
[10]Facchini, A., Module Theory. Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Progress in Mathematics, 167 (Birkhäuser, Basel–Boston–Berlin, 1998).Google Scholar
[11]Gómez Pardo, J. L., Dung, N. V. and Wisbauer, R., ‘Complete pure injectivity and endomorphism rings’, Proc. Amer. Math. Soc. 118 (1993), 1029–1034.CrossRefGoogle Scholar
[12]Gómez Pardo, J. L. and Guil Asensio, P. A., ‘Indecomposable decompositions of finitely presented pure-injective modules’, J. Algebra 192 (1997), 200–208.CrossRefGoogle Scholar
[13]Herzog, I., ‘Pure-injective envelopes’, J. Algebra Appl. 4 (2003), 397–402.CrossRefGoogle Scholar
[14]Nowak, S. and Simson, D., ‘Locally Dynkin quivers and hereditary coalgebras whose left comodules are direct sums of finite dimensional comodules’, Comm. Algebra 30 (2002), 455–476.CrossRefGoogle Scholar
[15]Osofsky, B. L., ‘Rings all of whose finitely generated modules are injective’, Pacific J. Math. 14 (1964), 645–650.CrossRefGoogle Scholar
[16]Osofsky, B. L. and Smith, P. F., ‘Cyclic modules whose quotients have all complement submodules direct summands’, J. Algebra 139 (1991), 342–354.CrossRefGoogle Scholar
[17]Prest, M., ‘Definable additive categories: purity and model theory’, Mem. Amer. Math. Soc. 210(987) (2011).Google Scholar
[18]Simson, D., ‘On pure-semisimple Grothendieck categories I’, Fund. Math. 100 (1978), 211–222.CrossRefGoogle Scholar
[19]Stenström, B., Rings of Quotients (Springer, Berlin–Heidelberg–New York, 1975).CrossRefGoogle Scholar
[20]Wisbauer, R., Foundations of Module and Ring Theory (Gordon and Breach, Reading, 1991).Google Scholar
[21]Xu, J., Flat Covers of Modules, Lecture Notes in Mathematics, 1634 (Springer, Berlin, 1996).CrossRefGoogle Scholar