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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 application, 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].
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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 and 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.
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.
This article generalizes joint work of the first author and I. Swanson to the $s$-multiplicity recently introduced by the second author. For $k$ a field and $X = [ x_{i,j}]$ a $m times n$-matrix of variables, we utilize Grobner bases to give a closed form the length $lambda( k[X] / (I_2(X) + mathfrak{m}^{ lceil sq rceil} + mathfrak{m}^{[q]} ))$ where $s in mathbf{Z}[p^{-1}]$, $q$ is a sufficiently large power of $p$, and $mathfrak{m}$ is the homogeneous maximal ideal of $k[X]$. This shows this length is always eventually a {it polynomial} function of $q$ for all $s$.
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.
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