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On (ℵ, U)-Coherence of Modules and Rings

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Abstract

Let U be a flat right R-module and ℵ an infinite cardinal number. A left R-module M is said to be (ℵ, U)-coherent if every finitely generated submodule of every finitely generated M-projective module in σ[M] is (ℵ, U)-finitely presented in σ[M]. It is proved under some additional conditions that a left R-module M is (ℵ, U)-coherent if and only if \( {\prod\nolimits_{i \in 1}^{\aleph } U } \) is M-flat as a right R-module if and only if the (ℵ, U)-coherent dimension of M is equal to zero. We also give some characterizations of left (ℵ, U)-coherent dimension of rings and show that the left ℵ-coherent dimension of a ring R is the supremum of (ℵ, U)-coherent dimensions of R for all flat right R-modules U.

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Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P. R. China

    Zhong Kui Liu

  2. Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Javed Ahsan

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  1. Zhong Kui Liu
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  2. Javed Ahsan
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Correspondence to Zhong Kui Liu.

Additional information

Research supported by National Natural Science Foundation of China (10171082) and by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, P. R. C.

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Liu, Z.K., Ahsan, J. On (ℵ, U)-Coherence of Modules and Rings. Acta Math Sinica 20, 105–114 (2004). https://doi.org/10.1007/s10114-003-0294-y

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  • DOI: https://doi.org/10.1007/s10114-003-0294-y

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