Abstract: Let R be a ring, n a fixed nonnegative integer. The concepts of (n, 0)-FI-injective and (n, 0)-FI-flat modules, and then give some characterizations of these modules over left n-coherent rings are introduced . In addition, we investigate the left and right n-FI-resolutions of R-modules by left (right) derived functors Extn(−,−) (Torn(−,−) ) over a left n-coherent ring, where n-FI stands for the categories of all (n, 0)- injective left R-modules. These modules together with the left or right derived functors are used to study the (n, 0)-injective dimensions of modules and rings.
Abstract: In this paper we study some properties of GC-projective, injective and flat modules, where C is a semidualizing module and we discuss some connections between GC-projective, injective and flat modules , and we consider these properties under change of rings such that completions of rings, Morita equivalences and the localizations.
Abstract: Let R be a ring and n a fixed positive integer, we
investigate the properties of n-strongly Gorenstein projective, injective
and flat modules. Using the homological theory , we prove that
the tensor product of an n-strongly Gorenstein projective (flat) right
R -module and projective (flat) left R-module is also n-strongly
Gorenstein projective (flat). Let R be a coherent ring ,we prove that
the character module of an n -strongly Gorenstein flat left R -module
is an n-strongly Gorenstein injective right R -module . At last, let
R be a commutative ring and S a multiplicatively closed set of R ,
we establish the relation between n -strongly Gorenstein projective
(injective , flat ) R -modules and n-strongly Gorenstein projective
(injective , flat ) S−1R-modules. All conclusions in this paper is
helpful for the research of Gorenstein dimensions in future.