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ON COHERENCE OF ENDOMORPHISM RINGS

  • HAI-YAN ZHU (a1) and NAN-QING DING (a2)

Abstract

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.

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Copyright

Corresponding author

For correspondence; e-mail: haiyzhu@gmail.com

Footnotes

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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).

Footnotes

References

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[1]Anderson, F. W. and Fuller, K. R., Rings and Categories of Modules (Springer, New York, 1974).
[2]Angeleri-Hügel, L., ‘Endocoherent modules’, Pacific J. Math. 212(1) (2003), 111.
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[4]Chase, S. U., ‘Direct products of modules’, Trans. Amer. Math. Soc. 97 (1960), 457473.
[5]Cheatham, T. J. and Stone, D. R., ‘Flat and projective character modules’, Proc. Amer. Math. Soc. 81(2) (1981), 175177.
[6]Colby, R. R., ‘Rings which have flat injective modules’, J. Algebra 35 (1975), 239252.
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[8]Enochs, E. E., ‘Injective and flat covers, envelopes and resolvents’, Israel J. Math. 39 (1981), 189209.
[9]Enochs, E. E. and Jenda, O. M. G., Relative Homological Algebra (Walter de Gruyter, Berlin, 2000).
[10]Göbel, R. and Trlifaj, J., Approximations and Endomorphism Algebras of Modules (Walter de Gruyter, Berlin, 2006).
[11]Gomez Pardo, J. L. and Hernandez, J. M., ‘Coherence of endomorphism rings’, Arch. Math. (Basel) 48(1) (1987), 4052.
[12]Jain, S., ‘Flat and FP-injectivity’, Proc. Amer. Math. Soc. 41 (1973), 437442.
[13]Rotman, J. J., An Introduction to Homological Algebra (Academic Press, New York, 1979).
[14]Stenström, B., ‘Coherent rings and FP-injective modules’, J. London Math. Soc. 2 (1970), 323329.
[15]Wakamatsu, T., ‘Tilting modules and Auslander’s Gorenstein property’, J. Algebra 275 (2004), 339.
[16]Wisbauer, R., Foundations of Module and Ring Theory (Gordon and Breach, Philadelphia, PA, 1991).
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ON COHERENCE OF ENDOMORPHISM RINGS

  • HAI-YAN ZHU (a1) and NAN-QING DING (a2)

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