Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T21:12:31.916Z Has data issue: false hasContentIssue false

ON KRULL–SCHMIDT FINITELY ACCESSIBLE CATEGORIES

Published online by Cambridge University Press:  01 April 2011

SEPTIMIU CRIVEI*
Affiliation:
Faculty of Mathematics and Computer Science, ‘Babeş-Bolyai’ University, Str. M. Kogălniceanu 1, 400084 Cluj-Napoca, Romania (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let 𝒞 be a finitely accessible additive category with products, and let (Ui)iI 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.

Type
Research Article
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), 10691082.CrossRefGoogle Scholar
[3]Bühler, T., ‘Exact categories’, Expo. Math. 28 (2010), 169.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), 379392.CrossRefGoogle Scholar
[6]Crawley-Boevey, W., ‘Locally finitely presented additive categories’, Comm. Algebra 22 (1994), 16411674.CrossRefGoogle Scholar
[7]Crivei, S., Năstăsescu, C. and Torrecillas, B., ‘On the Osofsky–Smith theorem’, Glasg. Math. J. 52A (2010), 6167.CrossRefGoogle Scholar
[8]Crivei, S., Prest, M. and Torrecillas, B., ‘Covers in finitely accessible categories’, Proc. Amer. Math. Soc. 138 (2010), 12131221.CrossRefGoogle Scholar
[9]Dung, N. V. and García, J. L., ‘Additive categories of locally finite representation type’, J. Algebra 238 (2001), 200238.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), 10291034.CrossRefGoogle Scholar
[12]Gómez Pardo, J. L. and Guil Asensio, P. A., ‘Indecomposable decompositions of finitely presented pure-injective modules’, J. Algebra 192 (1997), 200208.CrossRefGoogle Scholar
[13]Herzog, I., ‘Pure-injective envelopes’, J. Algebra Appl. 4 (2003), 397402.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), 455476.CrossRefGoogle Scholar
[15]Osofsky, B. L., ‘Rings all of whose finitely generated modules are injective’, Pacific J. Math. 14 (1964), 645650.CrossRefGoogle Scholar
[16]Osofsky, B. L. and Smith, P. F., ‘Cyclic modules whose quotients have all complement submodules direct summands’, J. Algebra 139 (1991), 342354.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), 211222.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