Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. In this paper, we introduce and study the notions of $S$-pure $S$-exact sequences and $S$-absolutely pure modules which extend the classical notions of pure exact seq
uences and absolutely pure modules. And then we characterize $S$-von Neumann regular rings and uniformly $S$-Noetherian rings using $S$-absolutely pure modules.
In this paper, we introduce and study the class of $phi$-$w$-flat modules which are generalizations of both $phi$-flat modules and $w$-flat modules. The $phi$-$w$-weak global dimension $phi$-$w$-w.gl.dim$(R)$ of a commutative ring $R$ is also introdu
ced and studied. We show that, for a $phi$-ring $R$, $phi$-$w$-w.gl.dim$(R)=0$ if and only if $w$-$dim(R)=0$ if and only if $R$ is a $phi$-von Neumann ring. It is also proved that, for a strongly $phi$-ring $R$, $phi$-$w$-w.gl.dim$(R)leq 1$ if and only if each nonnil ideal of $R$ is $phi$-$w$-flat, if and only if $R$ is a $phi$-PvMR, if and only if $R$ is a PvMR.
In this note, we show that a strongly $phi$-ring $R$ is a $phi$-PvMR if and only if any $phi$-torsion free $R$-module is $phi$-$w$-flat, if and only if any divisible module is nonnil-absolutely $w$-pure module, if and only if any $h$-divisible module
is nonnil-absolutely $w$-pure module, if and only if any finitely generated nonnil ideal of $R$ is $w$-projective.
In this note, we show that any epimorphism originating at a von Neumann regular ring (not necessary commutative) is a universal localization. As an application, we prove that the Telescope Conjecture holds for the unbounded derived categories of von Neumann regular rings (not necessary commutative).
Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. First, we introduce and study the $S$-projective dimensions and $S$-injective dimensions of $R$-modules, and then explore the $S$-global dimension $S$-gl.dim$(R)$ of
a commutative ring $R$ which is defined to be the supremum of $S$-projective dimensions of all $R$-modules. Finally, we investigated the $S$-global dimension of factor rings and polynomial rings.
Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called $S$-projective provided that the induced sequence $0rightarrow {rm Hom}_R(P,A)rightarrow {rm Hom}_R(P,B)rightarrow {rm Hom}_R(P,C)rightarrow 0$ is $S$-exact for any
$S$-short exact sequence $0rightarrow Arightarrow Brightarrow Crightarrow 0$. Some characterizations and properties of $S$-projective modules are obtained. The notion of $S$-semisimple modules is also introduced. A ring $R$ is called an $S$-semisimple ring provided that every free $R$-module is $S$-semisimple. Several characterizations of $S$-semisimple rings are provided by using $S$-semisimple modules, $S$-projective modules, $S$-injective modules and $S$-split $S$-exact sequences.
In this paper, we introduce and study the $S$-weak global dimension $S$-w.gl.dim$(R)$ of a commutative ring $R$ for some multiplicative subset $S$ of $R$. Moreover, commutative rings with $S$-weak global dimension at most $1$ are studied. Finally, we
investigated the $S$-weak global dimension of factor rings and polynomial rings.
Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $T$ is called uniformly $S$-torsion provided that $sT=0$ for some $sin S$. The notion of $S$-exact sequences is also introduced from the viewpoint of uniformity. An $R$-module $F
$ is called $S$-flat provided that the induced sequence $0rightarrow Aotimes_RFrightarrow Botimes_RFrightarrow Cotimes_RFrightarrow 0$ is $S$-exact for any $S$-exact sequence $0rightarrow Arightarrow Brightarrow Crightarrow 0$. A ring $R$ is called $S$-von Neumann regular provided there exists an element $sin S$ satisfies that for any $ain R$ there exists $rin R$ such that $sa=ra^2$. We obtain that a ring $R$ an $S$-von Neumann regular ring if and only if any $R$-module is $S$-flat. Several properties of $S$-flat modules and $S$-von Neumann regular rings are obtained.
Let $R$ be a commutative ring. If the nilpotent radical $Nil(R)$ of $R$ is a divided prime ideal, then $R$ is called a $phi$-ring. In this paper, we first distinguish the classes of nonnil-coherent rings and $phi$-coherent rings introduced by Bacem a
nd Ali [10], and then characterize nonnil-coherent rings in terms of $phi$-flat modules and nonnil-FP-injective modules. A $phi$-ring $R$ is called a $phi$-IF ring if any nonnil-injective module is $phi$-flat. We obtain some module-theoretic characterizations of $phi$-IF rings. Two examples are given to distinguish $phi$-IF rings and IF $phi$-rings.
In this paper, we introduce and study the class $S$-$mathcal{F}$-ML of $S$-Mittag-Leffler modules with respect to all flat modules. We show that a ring $R$ is $S$-coherent if and only if $S$-$mathcal{F}$-ML is closed under submodules. As an applicati
on, we obtain the $S$-version of Chase Theorem: a ring $R$ is $S$-coherent if and only if any direct product of $R$ is $S$-flat if and only if any direct product of flat $R$-modules is $S$-flat. Consequently, we provide an answer to the open question proposed by D. Bennis and M. El Hajoui [3].
