Narrow Results

In this paper, we prove that the Bass class with respect to a semidualizing bimodule C contains all FP-injective S-modules. We introduce the definition of C-FP-injective modules, and give some characterizations of right coherent rings in terms of the C-flat S-modules and C-FP-injective S^op-modules. We discuss when every S-module has an C-flat preenvelope which is epic (or monic). In addition, we investigate the left and right -resolutions of R-modules by left derived functors Ext_n(-, -) over a left Noetherian ring S. As applications, some new characterizations of left perfect rings are induced by these modules associated with C. A few classical results of these rings are obtained as corollaries.

We introduce and investigate in this paper a kind of Tate homology of modules over a commutative coherent ring based on Tate ℱ_C-resolutions, where C is a semidualizing module. We show firstly that the class of modules admitting a Tate ℱ_C-resolution is equal to the class of modules of finite 𝒢(ℱ_C)-projective dimension. Then an Avramov–Martsinkovsky type exact sequence is constructed to connect such Tate homology functors and relative homology functors. Finally, motivated by the idea of Sather–Wagstaff et al. [Comparison of relative cohomology theories with respect to semidualizing modules, Math. Z. 264 (2010) 571–600], we establish a balance result for such Tate homology over a Cohen–Macaulay ring with a dualizing module by using a good conclusion provided in [E. E. Enochs, S. E. Estrada and A. C. Iacob, Balance with unbounded complexes, Bull. London Math. Soc. 44 (2012) 439–442].

Let $C$ be a semidualizing $R$-module, where $R$ is a commutative ring. We first introduce the definition of $C$-cotorsion modules, and obtain the properties of $C$-cotorsion modules. As applications, we give some new characterizations for perfect rings. Second, we study the Foxby equivalences between the subclasses of the Auslander class and that of the Bass class with respect to $C$. Finally, we discuss $C$-cotorsion dimensions and investigate the transfer properties of strongly $C$-cotorsion dimensions under almost excellent extensions.

The notion of relative derived category with respect to a subcategory is introduced. A triangle-equivalence, which extends a theorem of Gao and Zhang [Gorenstein derived categories, J. Algebra323 (2010) 2041–2057] to the bounded below case, is obtained. Moreover, we interpret the relative derived functor ${\mathrm{\text{Ext}}}_{{𝒳}{𝒜}}(-,-)$ as the morphisms in such derived category and give two applications.

Let $R$ be a commutative ring. We define and study $f$-projective modules with respect to a semidualizing $R$-module $C$, which are called $C$–$f$-projective modules. As consequences, we characterize several rings such as $\mathrm{\Pi }$-coherent rings and Artinian rings using $C$–$f$-projective modules. Some known results are extended.

Let $R$ be a Cohen–Macaulay local ring. We prove that the $n$th syzygy module of a maximal Cohen–Macaulay $R$-module cannot have a semidualizing direct summand for every $n\ge 1$. In particular, it follows that $R$ is Gorenstein if and only if some syzygy of a canonical module of $R$ has a nonzero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen–Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.

Given an $R$-module $C$ and a class of $R$-modules ${𝒟}$ over a commutative ring $R$, we investigate the relationship between the existence of ${𝒟}$-envelopes (respectively, ${𝒟}$-covers) and the existence of $\mathrm{\text{Hom}}(C,{𝒟})$-envelopes or $C\otimes {𝒟}$-envelopes (respectively, $\mathrm{\text{Hom}}(C,{𝒟})$-covers or $C\otimes {𝒟}$-covers) of modules. As a consequence, we characterize coherent rings, Noetherian rings, perfect rings and Artinian rings in terms of envelopes and covers by $C$-projective, $C$-flat, $C$-injective and $C$-$FP$-injective modules, where $C$ is a semidualizing $R$-module.

Let $R$ be a commutative ring and let $I\ne 0$ be a proper ideal of $R$. In this paper, we study some algebraic and homological properties of a family of rings $R{(I)}_{a,b}$, with $a,b\in R$, that are obtained as quotients of the Rees algebra associated with the ring $R$ and the ideal $I$. Specially, we study when $R{(I)}_{a,b}$ is a von Neumann regular ring, a semisimple ring and a Gaussian ring. Also, we study the classical global and weak global dimensions of $R{(I)}_{a,b}$. Finally, we investigate some homological properties of $R{(I)}_{a,b}$-modules and we show that $R$ and $I$ are Gorenstein projective $R{(I)}_{a,b}$-modules, provided some special conditions.

Let $R$ be an arbitrary ring. We use a strict ${𝒲}{𝒳}$-resolution $X$ of a complex $K$ with finite $\widetilde{\#\mathrm{\text{-}}{𝒳}}$-projective dimension, where ${𝒳}$ denotes a subcategory of right $R$-modules closed under extensions and direct summands and admits an injective cogenerator ${𝒲}$, to define the $n$th relative cohomology functor ${\mathrm{\text{Ext}}}_{{𝒳}C}^n(K,-)$ as ${\mathrm{\text{H}}}^{-n}{\mathscr{H}}\mathrm{\text{om}}(X,-)$. If a complex $L$ has finite $\widetilde{\#\mathrm{\text{-}}{𝒴}}$-injective dimension, then one can use a dual argument to define a notion of a relative cohomology functor ${\mathrm{\text{Ext}}}_{{𝒞}Y}^n(-,L)$, where ${𝒴}$ is a subcategory of right $R$-modules closed under extensions and direct summands and admits a projective generator. Under several orthogonal conditions, we show that there exists an isomorphism ${\mathrm{\text{Ext}}}_{{{𝒳}C}^n}(K,L)\cong {\mathrm{\text{Ext}}}_{{𝒞}{𝒴}}^n(K,L)$ of relative cohomology groups for each $n\in \mathbb{Z}$. This result simultaneously encompasses a balance result of Holm on Gorenstein projective and injective modules, a balance result of Sather-Wagstaff, Sharif and White on Gorenstein projective and injective modules with respect to semidualizing modules, and a balance result of Liu on Gorenstein projective and injective complexes. In particular, as an application of this result, we extend the above balance result of Sather-Wagstaff, Sharif and White to the setting of complexes.

We contribute to the theory of G-dimension relative to a semidualizing module $C$, in connection to the properties of a module being totally $C$-reflexive, $C$-$k$-torsionless, and $C$-$k$-syzygy, where $k\ge 0$ is an integer. We extend several known results, and for an integer $q\ge 0$ we introduce the class of $C$-$q$-Gorenstein rings. We also consider $C$-duals and initiate the study of $C$-valued derivation modules over a local ring $R$ of low depth; if $C=R$ and $R$ is a three-dimensional Gorenstein domain, we find a bound for the number of generators and we propose a question when $C$ is general.

In this paper, we introduce and study G_C-flat complexes over a commutative Noetherian ring, where C is a semidualizing module. We prove that G_C-flat complexes are actually the complexes of G_C-flat modules. This complements a result of Yang and Liang. As an application, we get that every complex has a -cover, where is the class of G_C-flat complexes. We also give a characterization of complexes of modules in that are defined by Sather-Wagstaff, Sharif and White.