n-Strongly Gorenstein Projective, Injective and Flat Modules

Abstract

In this paper, we study the relation between m-strongly Gorenstein projective (resp. injective) modules and n-strongly Gorenstein projective (resp. injective) modules whenever m = n, and the homological behavior of n-strongly Gorenstein projective (resp. injective) modules. We introduce the notion of n-strongly Gorenstein flat modules. Then we study the homological behavior of n-strongly Gorenstein flat modules, and the relation between these modules and n-strongly Gorenstein projective (resp. injective)

$n$-strongly Gorenstein graded modules

“…The next result generalizes [27], Theorem 3.9, which gives a method how to construct a 1-SG-gr-injective module from n-SG-gr-injective modules.…”

“…In this section, we study the properties of n-strongly Gorenstein gr-injective and gr-projective modules. Some principal results of [9], [27] are generalized to n-strongly Gorenstein gr-injective or gr-projective modules. Definition 3.1.…”

“…Then the two ideals (X) and (Y ) of R are 2-SG-flat R-modules, but not 1-SG-flat by [27], Example 4.8, where (X) and (Y ) are the residue classes in R of X and Y respectively. By [27], Proposition 4.9, (X) + and (Y ) + are 2-SG-injective, and neither of them are 1-SG-injective by [25], Theorem 2.4. Since R may be viewed as a trivially graded ring, (X) + and (Y ) + are 2-SG-gr-injective, which are not 1-SG-gr-injective.…”

“…Furthermore, they discussed a generalization of strongly Gorenstein projective, injective and flat modules, named n-strongly Gorenstein projective, injective and flat modules, respectively. Zhao and Huang in [27] continued the study of homological behavior of the n-strongly Gorenstein projective, injective and flat modules.…”

“…In Section 2, we give some notation and collect some preliminary results. Then in Section 3, we give the definition of n-strongly Gorenstein gr-injective and gr-projective modules and generalize some principal results of [9], [27] to the n-strongly Gorenstein gr-injective modules. An example is given to show that n-strongly Gorenstein gr-injective modules need not be m-strongly Gorenstein gr-injective modules whenever n > m. The relations between the graded and the ungraded n-strongly Gorenstein injective modules are also discussed.…”