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 $R$ 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...

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;...

The author proved in 2018 that if $G$ is an open subset of a Hilbert space, $f_1,f_2:G\to ℝ$ continuous functions and $\omega$ a nontrivial modulus such that $f_1\le f_2$, $f_1$ is locally semiconvex with modulus $\omega$ and $f_2$ is locally semiconcave with modulus $\omega$, then there exists $f\in C_{\text{loc}}^{1,\omega }\left(G\right)$ such that $f_1\le f\le f_2$. This is a generalization of Ilmanen’s lemma (which deals with linear modulus and functions on an open subset of ${ℝ}^n$). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to $L^p$ spaces, $p\in [2,\infty )$. We...

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...

A nonzero $R$-module $M$ is atomic if for each two nonzero elements $a,b$ in $M$, both cyclic submodules $Ra$ and $Rb$ have nonzero isomorphic submodules. In this article it is shown that for an infinite $P$-space $X$, the factor rings $C(X,\mathbb{Z})/C_F(X,\mathbb{Z})$ and $C_c\left(X\right)/C_F\left(X\right)$ have no atomic ideals. This fact generalizes a result published in paper by A. Mozaffarikhah, E. Momtahan, A. R. Olfati and S. Safaeeyan (2020), which says that for an infinite set $X$, the factor ring ${\mathbb{Z}}^X/{\mathbb{Z}}^{\left(X\right)}$ has no atomic ideal. Another result is that for each infinite...

Let $n,d$ be two non-negative integers. A left $R$-module $M$ is called $(n,d)$-injective, if ${\mathrm{Ext}}^{d+1}(N,M)=0$ for every $n$-presented left $R$-module $N$. A right $R$-module $V$ is called $(n,d)$-flat, if ${\mathrm{Tor}}_{d+1}(V,N)=0$ for every $n$-presented left $R$-module $N$. A left $R$-module $M$ is called weakly $n$-$FP$-injective, if ${\mathrm{Ext}}^n(N,M)=0$ for every $(n+1)$-presented left $R$-module $N$. A right $R$-module $V$ is called weakly $n$-flat, if ${\mathrm{Tor}}_n(V,N)=0$ for every $(n+1)$-presented left $R$-module $N$. In this paper, we give some characterizations and properties of $(n,d)$-injective modules and $(n,d)$-flat modules in...

We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring $R$. Our focus is on homological properties of contracting endomorphisms of $R$, e.g., the Frobenius endomorphism when $R$ contains a field of positive characteristic. For instance, in this case, when $R$ is $F$-finite and $C$ is a semidualizing $R$-complex, we prove that the following conditions are equivalent: (i) $C$ is a dualizing $R$-complex; (ii) $C\sim 𝐑{\mathrm{Hom}}_R{(}^{n}R,C)$ for some $n>0$; (iii) ${\mathrm{G}}_C{\text{-dim}}^{n}R<\infty$ and $C$ is derived...

Let $R$ be a Noetherian ring, and $I$ and $J$ be two ideals of $R$. Let $S$ be a Serre subcategory of the category of $R$-modules satisfying the condition $C_I$ and $M$ be a $ZD$-module. As a generalization of the $S$-$\mathrm{depth}(I,M)$ and $\mathrm{depth}(I,J,M)$, the $S$-$\mathrm{depth}$ of $(I,J)$ on $M$ is defined as $S$-$\mathrm{depth}(I,J,M)=inf\{S$-$\mathrm{depth}(𝔞,M):𝔞\in \tilde{\mathrm{W}}(I,J)\}$, and some properties of this concept are investigated. The relations between $S$-$\mathrm{depth}(I,J,M)$ and $H_{I,J}^i\left(M\right)$ are studied, and it is proved that $S$-$\mathrm{depth}(I,J,M)=inf\{i:H_{I,J}^i\left(M\right)\notin S\}$, where $S$ is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology...

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Let $R$ be a prime ring of characteristic different from 2 and 3, $Q_r$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n\ge 1$ a fixed positive integer. Let $\alpha$ be an automorphism of the ring $R$. An additive map $D:R\to R$ is called an $\alpha$-derivation (or a skew derivation) on $R$ if $D\left(xy\right)=D\left(x\right)y+\alpha \left(x\right)D\left(y\right)$ for all $x,y\in R$. An additive mapping $F:R\to R$ is called a generalized $\alpha$-derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F\left(xy\right)=F\left(x\right)y+\alpha \left(x\right)D\left(y\right)$ for all $x,y\in R$. We prove...