Strongly Hopfian and Strongly Cohopfian Objects in the Category of Complexes of Left A-Modules

The object of this paper is the study of \emph{strongly hopfian}, \emph{strongly cohopfian}, \emph{quasi-injective}, \emph{quasi-projective}, \emph{Fitting} objects of the category of complexes of $A$-modules. In this paper we demonstrate the following results: a)If $C$ is a  strongly hopfian chain complex (respectively strongly cohopfian chain complex) and  $E$ a subcomplex which is direct summand  then $E$ and $C/E$ are both strongly Hopfian (respectively  strongly coHopfian) then $C$ is strongly Hopfian (respectively  strongly coHopfian). b)Given a chain complex $C$, if $C$ is quasi-injective and strongly-hopfian then $C$ is strongly cohopfian. c)Given a chain complex $C$, if $C$ is quasi-projective and strongly-cohopfian then $C$ is strongly hopfian. In conclusion the main result of this article is the following theorem: Any \emph{quasi-projective} and \emph{strongly-hopfian} or \emph{quasi-injective} and \emph{strongly-cohofian} chain complex of $A$-modules is a \emph{Fitting} chain complex.