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: Low-carbon economy means the energy conservation and emission reduction. How to measure and evaluate the regional low-carbon economy is an important problem which should be solved immediately. This paper proposed the eco-efficiency ratio based on the ecological efficiency to evaluate the current situation of the low-carbon economy in Jiangsu province and to analyze the efficiency of the low-carbon economy in Jiangsu and other provinces, compared both advantages and disadvantages. And then this paper put forward some advices for the government to formulate the correct development policy of low-carbon economy, to improve the technology innovation capacity and the efficiency of resource allocation.
Abstract: Let T,U and V be rings with identity and M be a unitary (T,U)-bimodule, N be a unitary (U, V )- bimodule, D be a unitary (T, V )-bimodule . We characterize homomorphisms and isomorphisms of the generalized matrix ring Γ = T M D 0 U N 0 0 V .
Abstract: We introduce the notion of strongly ω -Gorenstein modules, where ω is a faithfully balanced self-orthogonal module. This gives a common generalization of both Gorenstein projective (injective) modules and ω-Gorenstein modules. We investigate some characterizations of strongly ω -Gorenstein modules. Consequently, some properties under change of rings are obtained.
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.