Let $R$ be a ring. A subclass $𝒯$ of left $R$-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let $𝒯$ be a weak torsion class of left $R$-modules and $n$ a positive integer. Then a left $R$-module $M$ is called $𝒯$-finitely generated if there exists a finitely generated submodule $N$ such that $M/N\in 𝒯$; a left $R$-module $A$ is called $(𝒯,n)$-presented if there exists an exact sequence of left $R$-modules $$0⟶K_{n-1}⟶F_{n-1}⟶\cdots ⟶F_1⟶F_0⟶M⟶0$$ such that $F_0,\cdots ,F_{n-1}$ are finitely generated free and $K_{n-1}$ is $𝒯$-finitely generated;...

Let $I$ and $J$ be ideals of a Noetherian local ring $(R,𝔪)$ and let $M$ be a nonzero finitely generated $R$-module. We study the relation between the vanishing of $H_{I,J}^{dimM}\left(M\right)$ and the comparison of certain ideal topologies. Also, we characterize when the integral closure of an ideal relative to the Noetherian $R$-module $M/JM$ is equal to its integral closure relative to the Artinian $R$-module $H_{I,J}^{dimM}\left(M\right)$.

Let $R$ and $S$ be commutative rings with identity, $J$ be an ideal of $S$, $f:R\to S$ be a ring homomorphism, $M$ be an $R$-module, $N$ be an $S$-module, and let $\varphi :M\to N$ be an $R$-homomorphism. The amalgamation of $R$ with $S$ along $J$ with respect to $f$ denoted by $R{\bowtie }^{f}J$ was introduced by M. D’Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of $\left(R{\bowtie }^{f}J\right)$-module called the amalgamation of $M$ and $N$ along $J$ with respect to $\varphi$, and denoted by $M{\bowtie }^{\varphi }JN$. We study some homological properties of the $\left(R{\bowtie }^{f}J\right)$-module $M{\bowtie }^{\varphi }JN$. Among...

We investigate some properties of $n$-submodules. More precisely, we find a necessary and sufficient condition for every proper submodule of a module to be an $n$-submodule. Also, we show that if $M$ is a finitely generated $R$-module and $\sqrt{{\mathrm{Ann}}_R\left(M\right)}$ is a prime ideal of $R$, then $M$ has $n$-submodule. Moreover, we define the notion of $G.n$-submodule, which is a generalization of the notion of $n$-submodule. We find some characterizations of $G.n$-submodules and we examine the way the aforementioned notions are related...

Let $R$ be a commutative Noetherian ring and let $C$ be a semidualizing $R$-module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to $C$ which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every $G_C$-injective module $G$, the character module $G^{+}$ is $G_C$-flat, then the class ${\mathrm{𝒢ℐ}}_C\left(R\right)\cap {𝒜}_C\left(R\right)$ is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class ${\mathrm{𝒢ℐ}}_C\left(R\right)\cap {𝒜}_C\left(R\right)$...

Let $𝒯$ be a weak torsion class of left $R$-modules and $n$ a positive integer. A left $R$-module $M$ is called $(𝒯,n)$-injective if ${\mathrm{Ext}}_R^n(C,M)=0$ for each $(𝒯,n+1)$-presented left $R$-module $C$; a right $R$-module $M$ is called $(𝒯,n)$-flat if ${\mathrm{Tor}}_n^R(M,C)=0$ for each $(𝒯,n+1)$-presented left $R$-module $C$; a left $R$-module $M$ is called $(𝒯,n)$-projective if ${\mathrm{Ext}}_R^n(M,N)=0$ for each $(𝒯,n)$-injective left $R$-module $N$; the ring $R$ is called strongly $(𝒯,n)$-coherent if whenever $0\to K\to P\to C\to 0$ is exact, where $C$ is $(𝒯,n+1)$-presented and $P$ is finitely generated projective, then $K$ is $(𝒯,n)$-projective; the ring $R$ is called...

Let $(R,𝔪)$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${\mathrm{mAss}}_R(R/I)={\mathrm{Assh}}_R\left(I\right)$. It is shown that the $R$-module $H_I^{\mathrm{ht}\left(I\right)}\left(R\right)$ is $I$-cofinite if and only if $\mathrm{cd}(I,R)=\mathrm{ht}\left(I\right)$. Also we present a sufficient condition under which this condition the $R$-module $H_I^i\left(R\right)$ is finitely generated if and only if it vanishes.

A QTAG-module $M$ is an $\alpha$-module, where $\alpha$ is a limit ordinal, if $M/H_{\beta }\left(M\right)$ is totally projective for every ordinal $\beta <\alpha$. In the present paper $\alpha$-modules are studied with the help of $\alpha$-pure submodules, $\alpha$-basic submodules, and $\alpha$-large submodules. It is found that an $\alpha$-closed $\alpha$-module is an $\alpha$-injective. For any ordinal $\omega \le \alpha \le {\omega }_1$ we prove that an $\alpha$-large submodule $L$ of an ${\omega }_1$-module $M$ is summable if and only if $M$ is summable.

In this paper, we study some properties of $G_C$-flat $R$-modules, where $C$ is a semidualizing module over a commutative ring $R$ and we investigate the relation between the $G_C$-yoke with the $C$-yoke of a module as well as the relation between the $G_C$-flat resolution and the flat resolution of a module over $GF$-closed rings. We also obtain a criterion for computing the $G_C$-flat dimension of modules.

Let $R$ be a fusion ring and $R_{ℂ}:=R{\otimes }_{\mathbb{Z}}ℂ$ be the corresponding fusion algebra. We first show that the algebra $R_{ℂ}$ has only one left (right, two-sided) cell and the corresponding left (right, two-sided) cell module. Then we prove that, up to isomorphism, $R_{ℂ}$ admits a unique special module, which is 1-dimensional and given by the Frobenius-Perron homomorphism FPdim. Moreover, as an example, we explicitly determine the special module of the interpolated fusion algebra $R\left(\mathrm{PSL}(2,q)\right):=r\left(\mathrm{PSL}(2,q)\right){\otimes }_{\mathbb{Z}}ℂ$ up to isomorphism, where $r\left(\mathrm{PSL}\right(2,q\left)\right)$ is the...

Let $A$ be a CM-finite Artin algebra with a Gorenstein-Auslander generator $E$, $M$ be a Gorenstein projective $A$-module and $B={\mathrm{End}}_{A}M$. We give an upper bound for the finitistic dimension of $B$ in terms of homological data of $M$. Furthermore, if $A$ is $n$-Gorenstein for $2\le n<\infty$, then we show the global dimension of $B$ is less than or equal to $n$ plus the $B$-projective dimension of ${\mathrm{Hom}}_A(M,E).$ As an application, the global dimension of ${\mathrm{End}}_{A}E$ is less than or equal to $n$.

Let $(R,𝔪)$ be a commutative Noetherian local ring, $𝔞$ be an ideal of $R$ and $M$ a finitely generated $R$-module such that $𝔞M\ne M$ and $\mathrm{cd}(𝔞,M)-\mathrm{grade}(𝔞,M)\le 1$, where $\mathrm{cd}(𝔞,M)$ is the cohomological dimension of $M$ with respect to $𝔞$ and $\mathrm{grade}(𝔞,M)$ is the $M$-grade of $𝔞$. Let $D(-):={\mathrm{Hom}}_R(-,E)$ be the Matlis dual functor, where $E:=E(R/𝔪)$ is the injective hull of the residue field $R/𝔪$. We show that there exists the following long exact sequence $$\begin{array}{cc}\hfill 0⟶& H_{𝔞}^{n-2}\left(D\left(H_{𝔞}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^n\left(D\left(H_{𝔞}^n\left(M\right)\right)\right)⟶D\left(M\right)\\ \hfill ⟶& H_{𝔞}^{n-1}\left(D\left(H_{𝔞}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^{n+1}\left(D\left(H_{𝔞}^n\left(M\right)\right)\right)\\ \hfill ⟶& H_{𝔞}^n\left(D\left(H_{(x_1,...,x_{n-1})}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^n\left(D\left(H_{(}^{n-1}M\right)\right))⟶...,\end{array}$$ where $n:=\mathrm{cd}(𝔞,M)$ is a non-negative integer, $x_1,...,x_{n-1}$ is a regular sequence in $𝔞$ on $M$ and, for an $R$-module $L$, $H_{𝔞}^i\left(L\right)$ is the $i$th local cohomology module...