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Register a new user$S$-absolutely pure modules
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Xiaolei Zhang
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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 sequences and absolutely pure modules. And then we characterize $S$-von Neumann regular rings and uniformly $S$-Noetherian rings using $S$-absolutely pure modules.
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We show that the property of a standard graded algebra R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module corresponding to any degree sequence of length at most depth(R). We also give a relation in terms of graded Betti numbers, called the Herzog-Kuhl equations, for a pure R-module M to satisfy the condition dim(R) - depth(R) = dim(M) - depth(M). When R is Cohen-Macaulay, we prove an analogous result characterizing all graded Cohen-Macaulay R-modules.
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.
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.
We study absolutely Koszul algebras, Koszul algebras with the Backelin-Roos property and their behavior under standard algebraic operations. In particular, we identify some Veronese subrings of polynomial rings that have the Backelin-Roos property and conjecture that the list is indeed complete. Among other things, we prove that every universally Koszul ring defined by monomials has the Backelin-Roos property.
Let $A$ be a Noetherian flat $K[t]$-algebra, $h$ an integer and let $N$ be a graded $K[t]$-module, we introduce and study $N$-fiber-full up to $h$ $A$-modules. We prove that an $A$-module $M$ is $N$-fiber-full up to $h$ if and only if $mathrm{Ext}^i_A(M, N)$ is flat over $K[t]$ for all $ile h-1$. And we show some applications of this result extending the recent result on squarefree Grobner degenerations by Conca and Varbaro.
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