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Existence of Covers and Envelopes of a Left Orthogonal Class and Its Right Orthogonal Class of Modules

Author

Listed:
  • Arunachalam Umamaheswaran
  • Ramalingam Udhayakumar
  • Chelliah Selvaraj
  • Kandhasamy Tamilvanan
  • Masho Jima Kabeto
  • Ralf Meyer

Abstract

In this paper, we investigate the notions of X⊥-projective, X-injective, and X-flat modules and give some characterizations of these modules, where X is a class of left modules. We prove that the class of all X⊥-projective modules is Kaplansky. Further, if the class of all X-injective R-modules is contained in the class of all pure projective modules, we show the existence of X⊥-projective covers and X-injective envelopes over a X⊥-hereditary ring. Further, we show that a ring R is Noetherian if and only if W-injective R-modules coincide with the injective R-modules. Finally, we prove that if W⊆S, every module has a W-injective precover over a coherent ring, where W is the class of all pure projective R-modules and S is the class of all fp−Ω1-modules.

Suggested Citation

  • Arunachalam Umamaheswaran & Ramalingam Udhayakumar & Chelliah Selvaraj & Kandhasamy Tamilvanan & Masho Jima Kabeto & Ralf Meyer, 2022. "Existence of Covers and Envelopes of a Left Orthogonal Class and Its Right Orthogonal Class of Modules," Journal of Mathematics, Hindawi, vol. 2022, pages 1-12, October.
    • Handle: RePEc:hin:jjmath:8072134
    DOI: 10.1155/2022/8072134
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