arXiv subjects

A. -J. Taherizadeh

Publications and source records attributed to A. -J. Taherizadeh.

5 recordsLinked to original sources

arXiv2016

Codualizing modules

Let $(R, \mathfrak{m})$ be a Noetherian local ring. In this paper, we introduce a dual notion for dualizing modules, namely codualizing modules. We study the basic properties of codualizing modules and use them to establish an equivalence between the category of noetherian modules of finite projective dimension and the category of artinian modules of finite projective dimension. Next, we give some applications of codualizing modules. Finally, we present a mixed identity involving quasidualizing module that characterize the codualizing module. As an application, we obtain a necessary and sufficient condition for $R$ to be Gorenstein.

math.AC↗

arXiv2016

Tensor Product of $C$-Injective Modules

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, we are concerned with the tensor and torsion product of $C$-injective modules. Firstly, it is shown that the tensor product of any two $C$-injective $R$-modules is $C$-injective if and only if the injective hull of $C$ is $C$-flat. Secondly, it is proved that $C$ is a pointwise dualizing $R$-module if and only if $Tor^R_i(M,N)$ is $C$-injective for all $C$-injective $R$-modules $M$ and $N$, and all $ i \geq 0$. These results recover the celebrated theorems of Enochs and Jenda \cite{EJ2}.

math.AC↗

arXiv2016

Dual of Bass numbers and dualizing modules

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using relative homological dimensions with respect to $C$, we impose various conditions on $C$ to be dualizing. First, we show that $C$ is dualizing if and only if there exists a Cohen-Macaulay $R$-module of type 1 and of finite G$ _C $-dimension. This result extends Takahashi \cite[Theorem 2.3]{T} as well as Christensen \cite[Proposition 8.4]{C}. Next, as a generalization of Xu \cite[Theorem 3.2]{X2}, we show that $C$ is dualizing if and only if for an $R$-module $M$, the necessary and sufficient condition for $M$ to be $C$-injective is that $ π_i(\fp , M) = 0 $ for all $ \fp \in \Spec(R) $ and all $ i \neq \h(\fp) $, where $ π_i $ is the invariant dual to the Bass numbers defined by E.Enochs and J.Xu \cite{EX}. We use the later result to give an explicit structure of the minimal flat resolution of $ \H_{\fm}^d(R) $, where $ (R, \fm) $ is a $ d $-dimensional Cohen-Macaulay local ring possessing a canonical module. As an application, we compute the torsion product of these local cohomology modules.

math.AC↗

arXiv2016

Characterizing local rings via perfect and coperfect modules

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using the classes $ \mathcal{P}_C $ and $ \mathcal{I}_C $, we extend the notions of perfect and coperfect modules introduced by D.Rees \cite{R} and O.Jenda \cite{J1}. First, we study the basic properties of these modules and relations between them. Next, we characterize local rings in terms of the existence of special perfect (resp. coperfect) modules.

math.AC↗

arXiv2015

Some new Subcategories and homological dimensions related to Semidualizing modules

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using the semidualizing modules, we define and study new classes of modules and homological dimensions and investigate the relations between them. In parallel, we obtain some necessary and sufficient condition for $ C $ to be dualizing.

math.AC↗