Strongly Hopfian Objets and Strongly Cohopfian Objets in the Categories of Agr(a − Mod) and Comp(agr(a − Mod))

Abstract

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); 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. 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