Covers, precovers, and purity. (English) Zbl 1189.16007

A class \(\mathbb{F}\) of modules is (pre)covering if each module \(M\) has an \(\mathbb{F}\)-(pre)cover \(F\to M\) (\(F\in\mathbb{F}\)). If a class \(\mathbb{F}\) is closed under pure quotient modules, then the following conditions are equivalent: (i) \(\mathbb{F}\) is closed under set indexed direct sums; (ii) \(\mathbb{F}\) is precovering; (iii) \(\mathbb{F}\) is covering.
The pair of classes \((\mathbb{F},\mathbb{G})\) is called cotorsion pair if \(\mathbb{F}^\perp=\mathbb{G}\) and \(\mathbb{F}={^\perp\mathbb{G}}\), where \(\mathbb{F}^\perp=\text{Ker\,Ext}^1(\mathbb{F},-)\), \(^\perp\mathbb{G}=\text{Ker\,Ext}^1(-,\mathbb{G})\). If a class \(\mathbb{F}\) contains the ground ring \(R\) and is closed under extensions, set indexed direct sums, pure submodules and pure quotient modules, then \((\mathbb{F},\mathbb{F}^\perp)\) is a perfect cotorsion pair (in particular, \(\mathbb{F}\) is covering and \(\mathbb{F}^\perp\) is enveloping).
As applications some concrete classes of modules are considered: the kernels of homological functors, the torsion free classes for torsion pairs.


16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
16D40 Free, projective, and flat modules and ideals in associative algebras
16D90 Module categories in associative algebras
Full Text: Euclid


[1] D. D. Adams, Absolutely pure modules , Thesis, University of Kentucky, 1978.
[2] L. Angeleri Hügel, A. Tonolo and J. Trlifaj, Tilting preenvelopes and cotilting precovers , Algebr. Represent. Theory 4 (2001), 155–170. · Zbl 0999.16007
[3] L. Bican, R. El Bashir and E. E. Enochs, All modules have flat covers , Bull. London Math. Soc. 33 (2001), 385–390. · Zbl 1029.16002
[4] L. Bican and B. Torrecillas, On covers , J. Algebra 236 (2001), 645–650. · Zbl 0973.16002
[5] L. Bican and B. Torrecillas, Precovers , Czechoslovak Math. J. 53 (2003), 191–203. · Zbl 1016.16003
[6] S. E. Dickson, A torsion theory for Abelian categories , Trans. Amer. Math. Soc. 121 (1966), 223–235. · Zbl 0138.01801
[7] P. C. Eklof and J. Trlifaj, Covers induced by Ext , J. Algebra 231 (2000), 640–651. · Zbl 0981.16010
[8] R. El Bashir, Covers and directed colimits , Algebr. Represent. Theory 9 (2006), 423–430. · Zbl 1114.16002
[9] E. E. Enochs, Injective and flat covers, envelopes and resolvents , Israel J. Math. 39 (1981), 189–209. · Zbl 0464.16019
[10] E. E. Enochs and O. M. G. Jenda, Relative homological algebra , de Gruyter Exp. Math., vol. 30, de Gruyter, Berlin, 2000. · Zbl 0952.13001
[11] E. E. Enochs and J. A. López-Ramos, Kaplansky classes , Rend. Sem. Mat. Univ. Padova 107 (2002), 67–79.
[12] C. U. Jensen and H. Lenzing, Model theoretic algebra , Algebra Logic Appl., vol. 2, Gordon and Breach, New York, 1989. · Zbl 0728.03026
[13] H. Krause and Ø. Solberg, Applications of cotorsion pairs , J. London Math. Soc. (2) 68 (2003), 631–650. · Zbl 1061.16006
[14] K. R. Pinzon, Absolutely pure modules , Thesis, University of Kentucky, 2005.
[15] J. Rada and M. Saorín, Rings characterized by (pre)envelopes and (pre)covers of their modules , Comm. Algebra 26 (1998), 899–912. · Zbl 0908.16003
[16] L. Salce, Cotorsion theories for Abelian groups , Conference on Abelian Groups and their Relationship to the Theory of Modules (Istituto Nazionale di Alta Matematica Francesco Severi (INDAM), Rome, 12–16 December 1977), Sympos. Math., vol. XXIII, Academic Press, London, 1979, pp. 11–32. · Zbl 0426.20044
[17] M. Teply, Torsionfree injective modules , Pacific J. Math. 28 (1969), 441–453. · Zbl 0172.32001
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