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ON COHERENCE OF ENDOMORPHISM RINGS

Part of: Modules, bimodules and ideals

Published online by Cambridge University Press:  26 January 2010

HAI-YAN ZHU  and
NAN-QING DING
HAI-YAN ZHU*
Affiliation:
Department of Mathematics, Zhejiang University of Technology, Zhejiang 310023, PR China (email: [email protected])
NAN-QING DING
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let R be a ring and U a left R-module with S=End(RU). The aim of this paper is to characterize when S is coherent. We first show that a left R-module F is TU-flat if and only if HomR(U,F) is a flat left S-module. This removes the unnecessary hypothesis that U is Σ-quasiprojective from Proposition 2.7 of Gomez Pardo and Hernandez [‘Coherence of endomorphism rings’, Arch. Math. (Basel)48(1) (1987), 40–52]. Then it is shown that S is a right coherent ring if and only if all direct products of TU-flat left R-modules are TU-flat if and only if all direct products of copies of RU are TU-flat. Finally, we prove that every left R-module is TU-flat if and only if S is right coherent with wD(S)≤2 and US is FP-injective.

Keywords

TU-flat modulecoherent ringendomorphism ring

MSC classification

Secondary: 16D40: Free, projective, and flat modules and ideals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 2 , April 2010 , pp. 186 - 194
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

This work was partially supported by the National Science Foundation of China (Grant No. 10771096) and the Natural Science Foundation of Jiangsu Province of China (Grant No. BK2008365).

References

[1]Anderson, F. W. and Fuller, K. R., Rings and Categories of Modules (Springer, New York, 1974).Google Scholar
[2]Angeleri-Hügel, L., ‘Endocoherent modules’, Pacific J. Math. 212(1) (2003), 1–11.CrossRefGoogle Scholar
[3]Beligiannis, A. and Reiten, I., Homological and Homotopical Aspects of Torsion Theories, Memoirs of the American Mathematical Society, 188 (American Mathematical Society, Providence, RI, 2007).CrossRefGoogle Scholar
[4]Chase, S. U., ‘Direct products of modules’, Trans. Amer. Math. Soc. 97 (1960), 457–473.CrossRefGoogle Scholar
[5]Cheatham, T. J. and Stone, D. R., ‘Flat and projective character modules’, Proc. Amer. Math. Soc. 81(2) (1981), 175–177.Google Scholar
[6]Colby, R. R., ‘Rings which have flat injective modules’, J. Algebra 35 (1975), 239–252.CrossRefGoogle Scholar
[7]Colby, R. R. and Fuller, K. R., Equivalence and Duality for Module Categories (with Tilting and Cotilting for Rings), Cambridge Tracts in Mathematics, 161 (Cambridge University Press, Cambridge, 2004).Google Scholar
[8]Enochs, E. E., ‘Injective and flat covers, envelopes and resolvents’, Israel J. Math. 39 (1981), 189–209.Google Scholar
[9]Enochs, E. E. and Jenda, O. M. G., Relative Homological Algebra (Walter de Gruyter, Berlin, 2000).Google Scholar
[10]Göbel, R. and Trlifaj, J., Approximations and Endomorphism Algebras of Modules (Walter de Gruyter, Berlin, 2006).Google Scholar
[11]Gomez Pardo, J. L. and Hernandez, J. M., ‘Coherence of endomorphism rings’, Arch. Math. (Basel) 48(1) (1987), 40–52.Google Scholar
[12]Jain, S., ‘Flat and FP-injectivity’, Proc. Amer. Math. Soc. 41 (1973), 437–442.CrossRefGoogle Scholar
[13]Rotman, J. J., An Introduction to Homological Algebra (Academic Press, New York, 1979).Google Scholar
[14]Stenström, B., ‘Coherent rings and FP-injective modules’, J. London Math. Soc. 2 (1970), 323–329.CrossRefGoogle Scholar
[15]Wakamatsu, T., ‘Tilting modules and Auslander’s Gorenstein property’, J. Algebra 275 (2004), 3–39.Google Scholar
[16]Wisbauer, R., Foundations of Module and Ring Theory (Gordon and Breach, Philadelphia, PA, 1991).Google Scholar