We apply the Auslander-Buchweitz approximation theory to show that the Iyama and Yoshinos subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a
result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitzs triangle equivalence from Iwanaga-Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitzs triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga-Gorenstein rings and Gorenstein algebras
Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=left(begin{smallmatrix} A & 0 U & B end{smallmatrix}right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over $T$, and discuss
when a left $T$-module is strongly Gorenstein projective or strongly Gorenstein injective module.
Let $R$ be a commutative noetherian ring with a semi-dualizing module $C$. The Auslander categories with respect to $C$ are related through Foxby equivalence: $xymatrix@C=50pt{mathcal {A}_C(R) ar@<0.4ex>[r]^{Cotimes^{mathbf{L}}_{R} -} & mathcal {B}_C
(R) ar@<0.4ex>[l]^{mathbf{R}mathrm{Hom}_{R}(C, -)}}$. We firstly intend to extend the Foxby equivalence to Cartan-Eilenberg complexes. To this end, C-E Auslander categories, C-E $mathcal{W}$ complexes and C-E $mathcal{W}$-Gorenstein complexes are introduced, where $mathcal{W}$ denotes a self-orthogonal class of $R$-modules. Moreover, criteria for finiteness of C-E Gorenstein dimensions of complexes in terms of resolution-free characterizations are considered.
The notion of an $mathcal{L}$ complex (for a given class of $R$-modules $mathcal{L}$) was introduced by Gillespie: a complex $C$ is called $mathcal{L}$ complex if $C$ is exact and $Z_{i}(C)$ is in $mathcal{L}$ for all $iin mathbb{Z}$. Let $widetilde{
mathcal{L}}$ stand for the class of all $mathcal{L}$ complexes. In this paper, we give sufficient condition on a class of $R$-modules such that every complex has a special $widetilde{mathcal{L}}$-precover (resp., $widetilde{mathcal{L}}$-preenvelope). As applications, we obtain that every complex has a special projective precover and a special injective preenvelope, over a coherent ring every complex has a special FP-injective preenvelope, and over a noetherian ring every complex has a special $widetilde{mathcal{GI}}$-preenvelope, where $mathcal{GI}$ denotes the class of Gorenstein injective modules.
A left $R$-module $M$ is called two-degree Ding projective if there exists an exact sequence $...longrightarrow D_{1}longrightarrow D_{0}longrightarrow D_{-1}longrightarrow D_{-2}longrightarrow...$ of Ding projective left $R$-modules such that $Mcong
ker (D_{0}longrightarrow D_{-1})$ and $Hom_{R} (-, F)$ leaves the sequence exact for any flat (or Gorenstein flat) left $R$-module $F$. In this paper, we show that the two-degree Ding projective modules are nothing more than the Ding projective modules.
In this paper, we define and study a notion of Ding projective dimension for complexes of left modules over associative rings. In particular, we consider the class of homologically bounded below complexes of left R-modules, and show that Ding projective dimension has a nice functorial description.
In this paper, we first introduce $mathcal {W}_F$-Gorenstein modules to establish the following Foxby equivalence: $xymatrix@C=80pt{mathcal {G}(mathcal {F})cap mathcal {A}_C(R) ar@<0.5ex>[r]^{Cotimes_R-} & mathcal {G}(mathcal {W}_F) ar@<0.5ex>[l]^{te
xtrm{Hom}_R(C,-)}} $ where $mathcal {G}(mathcal {F})$, $mathcal {A}_C(R) $ and $mathcal {G}(mathcal {W}_F)$ denote the class of Gorenstein flat modules, the Auslander class and the class of $mathcal {W}_F$-Gorenstein modules respectively. Then, we investigate two-degree $mathcal {W}_F$-Gorenstein modules. An $R$-module $M$ is said to be two-degree $mathcal {W}_F$-Gorenstein if there exists an exact sequence $mathbb{G}_bullet=indent ...longrightarrow G_1longrightarrow G_0longrightarrow G^0longrightarrow G^1longrightarrow...$ in $mathcal {G}(mathcal {W}_F)$ such that $M cong$ $im(G_0rightarrow G^0) $ and that $mathbb{G}_bullet$ is Hom$_R(mathcal {G}(mathcal {W}_F),-)$ and $mathcal {G}(mathcal {W}_F)^+otimes_R-$ exact. We show that two notions of the two-degree $mathcal {W}_F$-Gorenstein and the $mathcal {W}_F$-Gorenstein modules coincide when R is a commutative GF-closed ring.
An important instance of Rota-Baxter algebras from their quantum field theory application is the ring of Laurent series with a suitable projection. We view the ring of Laurent series as a special case of generalized power series rings with exponents
in an ordered monoid. We study when a generalized power series ring has a Rota-Baxter operator and how this is related to the ordered monoid.
