$n$-strongly Gorenstein graded modules

summary:Let $R$ be a graded ring and $n\geq 1$ an integer. We introduce and study $n$-strongly Gorenstein gr-projective, gr-injective and gr-flat modules. Some examples are given to show that $n$-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules need not be $m$-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules whenever $n>m$. Many properties of the $n$-strongly Gorenstein gr-injective and gr-flat modules are discussed, some known results are generalized. Then we investigate the relations between the graded and the ungraded $n$-strongly Gorenstein injective (or flat) modules. In addition, the connections between the $n$-strongly Gorenstein gr-projective, gr-injective and gr-flat modules are considered