Hopfian and Cohopfian Objects in the Categories of Gr(A - Mod) and COMP(Gr(A - Mod))

Abstract

We study in this work the notions of hopficity and cohopficity in the categories AGr(A - Mod) and COMP(AGr(A - Mod)) of associate complex to a graded left A-module and we show that: 1. Let M a graded left A-module, N a graded submodule of M, M_* be a complex associate to M. Suppose that M_* be a quasi-projective and N be a completely invariant and essential sub-complex of M_* associate to N. Then N_* is cohopfian if, and only, if M_* is cohopfian. 2. Let M a graded left A-module, N a graded submodule of M, M_* quasi-injective and M_* a completely invariant and superfluous sub-complex of M_*. Then M_* is cohopfian if, and only, if M_*=M_* is cohopfian.