Keywords:-

Keywords: Graded rings, graded module, category, sequence complex, chains complex, strongly cohopfian objects, and strongly hopfian object.

Article Content:-

Abstract

  1. Let M be a graded leftA-module and M∗ the associate complex of M. Then :
    • If M∗ is noetherian (resp. artinian) then M∗ is strongly hopfian (resp. strongly cohopfian);
    • If M∗ is strongly hopfian (resp. cohopfian), then M∗ is hopfian (resp. cohopfian);
  2. Let M be a graded leftA-module, M∗ the associate complex of M, N a submodule of M, and N∗ fully invariant subcomplex of M∗. Then :

If N∗ and M∗/N∗ strongly hopfian, then M∗ is strongly hopfian.

  1. Let M a graded leftA-module, N a submodule of M and M∗ the associate complex of M. Then :
    • if all subcomplex of M∗ is cohopfian, then M∗ is cohopfian.
    • if M∗/N∗ is strongly hopfian, then M∗ is strongly hopfian

References:-

References

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BALDE, S. A., BEN MAAOUIA, M. B. F., & CHBIH, A. O. (2023). Strongly Hopfian Objets and Strongly Cohopfian Objets in the Categories of AGr(A − Mod) and COMP(AGr(A − Mod)). International Journal Of Mathematics And Computer Research, 11(06), 3479-3493. https://doi.org/10.47191/ijmcr/v11i6.03