×

A new version of a theorem of Kaplansky. (English) Zbl 1446.13009

A well-known theorem of Kaplansky states that any projective module is a direct sum of countably generated modules [I. Kaplansky, Ann. Math. (2) 68, 372–377 (1958; Zbl 0083.25802)], which is equivalent to saying that every projective module can be filtered by countably generated projective modules. In order to get the \(w\)-analog of this theorem, the authors introduced the concepts of \(w\)-split modules and \(w\)-countably generated modules, where \(w\) is a hereditary torsion theory for modules over a commutative ring. The following is the main theorem of the paper.
Main Theorem. Let \(M\) be a \(w\)-projective \(w\)-module. Then
(1)
\(M\) can be filtered by countably generated \(w\)-split modules.
(2)
\(M\) can be filtered by \(w\)-countably generated \(w\)-projective modules.
Reviewer: Hwankoo Kim (Asan)

MSC:

13C13 Other special types of modules and ideals in commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
13D30 Torsion theory for commutative rings

Citations:

Zbl 0083.25802
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Enochs, E.; Estrada, S.; Iacob, A., Cotorsion pairs, model structures and homotopy categories, Houston J. Math, 40, 43-61 (2014) · Zbl 1304.18037
[2] Estrada, S.; Asensio, P. A. G.; Odabaş, S., A lazard-like theorem for quasi-coherent sheaves, Algebr. Represent. Theor., 16, 4, 1193-1205 (2013) · Zbl 1278.14026 · doi:10.1007/s10468-012-9353-3
[3] Glaz, S.; Vasconcelos, W. V., Flat ideals II, Manuscripta Math, 22, 4, 325-341 (1977) · Zbl 0367.13002 · doi:10.1007/BF01168220
[4] Göbel, R.; Trlifaj, J., Approximations and Endomorphism Algebras of Modules, 41 (2006), Berlin: Walter de Gruyter GmbH & Co, KG, Berlin · Zbl 1121.16002
[5] Hedstrom, J. R.; Houston, E. G., Some remarks on star-operations, J. Pure Appl. Algebra, 18, 37-44 (1980) · Zbl 0462.13003 · doi:10.1016/0022-4049(80)90114-0
[6] Hill, P., The third axiom of countability for abelian groups, Proc. Amer. Math. Soc, 82, 3, 347-350 (1981) · Zbl 0467.20041 · doi:10.1090/S0002-9939-1981-0612716-0
[7] Kaplansky, I., Projective modules, Ann. of Math, 68, 2, 372-377 (1958) · Zbl 0083.25802 · doi:10.2307/1970252
[8] Kaplansky, I., Commutative Rings (1974), Chicago: The University of Chicago Press, Chicago · Zbl 0296.13001
[9] Osofsky, B. L., Projective dimension of “nice” directed unions, J. Pure Appl. Algebra, 13, 2, 179-219 (1978) · Zbl 0392.16023 · doi:10.1016/0022-4049(78)90008-7
[10] Rotman, J. J., An Introduction to Homological Algebra (1979), New York: Academic Press, New York · Zbl 0441.18018
[11] Šťovíček, J.; Trlifaj, J., Generalized Hill lemma, Kaplansky theorem for cotorsion pairs and some applications, Rocky Mountain J. Math, 39, 1, 305-324 (2009) · Zbl 1173.16004 · doi:10.1216/RMJ-2009-39-1-305
[12] Wang, F., On w-projective modules and w-flat modules, Algebra Colloq, 4, 111-120 (1997) · Zbl 0883.13009
[13] Wang, F.; Kim, H., Two generalizations of projective modules and their applications, J. Pure Appl. Algebra, 219, 6, 2099-2123 (2015) · Zbl 1337.13009 · doi:10.1016/j.jpaa.2014.07.025
[14] Wang, F.; Kim, H., Foundations of Commutative Rings and Their Modules,, 22 (2016), Singapore: Springer, Singapore · Zbl 1367.13001
[15] Yin, H.; Wang, F.; Zhu, X.; Chen, Y., w-modules over commutative rings, J. Korean Math. Soc, 48, 1, 207-222 (2011) · Zbl 1206.13005 · doi:10.4134/JKMS.2011.48.1.207
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.