In the present paper, modules over integral domains and principal ideal domains that are proper essential extensions of some submodules are classified. We introduce a new class of modules that we call $\mathrm{SM}$ modules and show that the class of Artinian modules, locally finite modules, and modules of finite lengths are all proper subclasses of SM modules. We also show that non-semisimple $\mathrm{SM}$ modules possess essential socles. Further, we show that non-semieimple modules over integral domains with nonempty torsion-free parts do not possess essential socles.
In this paper, we discuss some properties on Lucas modules. In details, we show that direct and inverse limits of Lucas modules are Lucas modules, and every $R$-module has a Lucas envelope and a Lucas cover. Moreover, some properties of direct and inverse limits of Lucas modules and some constructions and the unique mapping properties of Lucas envelopes and Lucas covers are investigated.
Let $R=k[x,y]$ be a polynomial ring over a field $k$ of prime characteristic $p$ and let $E$ denote the injective hull of $k$ (which is isomorphic to $H^2_{(x,y)}(R)$). We prove that $E$ is not an injective object in the category of graded $F$-modules over $R$. This answers in the negative a question raised by Lyubeznik-Singh-Walther.
In this paper, the notions of nonnil-injective modules and nonnil-FP-injective modules are introduced and studied. Especially, we show that a $ϕ$-ring $R$ is an integral domain if and only if any nonnil-injective (resp., nonnil-FP-injective) module $R$-module is injective (resp., FP-injective). Some new characterizations of $ϕ$-von Neumann regular rings, nonnil-Notherian rings and nonnil-coherent rings are given. We finally characterize $ϕ$-Dedekind rings and $ϕ$-\Prufer\ rings in terms of $ϕ$-flat modules, nonnil-injective modules and nonnil-FP-injective modules.
School of Mathematics and Statistics, Tianshui Normal University（天水师范学院数学与统计学院）；School of Mathematics and Physics, Southwest University of Science and Technology（西南科技大学数学与物理学院）
Let $R$ be a commutative ring. If the nilpotent radical $Nil(R)$ of $R$ is a divided prime ideal, then $R$ is called a $ϕ$-ring. In this paper, we first distinguish the classes of nonnil-coherent rings and $ϕ$-coherent rings introduced by Bacem and Ali [10], and then characterize nonnil-coherent rings in terms of $ϕ$-flat modules and nonnil-FP-injective modules. A $ϕ$-ring $R$ is called a $ϕ$-IF ring if any nonnil-injective module is $ϕ$-flat. We obtain some module-theoretic characterizations of $ϕ$-IF rings. Two examples are given to distinguish $ϕ$-IF rings and IF $ϕ$-rings.
Let $R$ be a commutative ring with unity, $M$ be a unitary $R$-module and $G$ a finite abelian group (viewed as a $\mathbb{Z}$-module). The main objective of this paper is to study properties of mod-annihilators of $M$. For $x \in M$, we study the ideals $[x : M] =\{r\in R | rM\subseteq Rx\}$ of $R$ corresponding to mod-annihilator of $M$. We investigate that when $[x : M]$ is an essential ideal of $R$. We prove that arbitrary intersection of essential ideals represented by mod-annihilators is an essential ideal. We observe that $[x : M]$ is injective if and only if $R$ is non-singular and the…
In this paper we study a long-standing conjecture of Huneke and Wiegand which is concerned with the torsion submodule of certain tensor products of modules over one-dimensional local domains. We utilize Hochster's theta invariant and show that the conjecture is true for two periodic modules. We also make use of a result of Orlov and formulate a new condition which, if true over hypersurface rings, forces the conjecture of Huneke and Wiegand to be true over complete intersection rings of arbitrary codimension. Along the way we investigate the interaction between the vanishing of Tate (co)homolo…
In this paper, we study the class of modules have the property that every pure submodule is essential in a direct summand. These modules are termed as pure extending modules which is a proper generalisation of extending modules. Examples and counterexamples are given. We study some properties of pure extending modules and characterize regular ring, semisimple ring, local ring and PDS ring in terms of pure extending modules.
This paper is concerned with S-co-m modules which are a generalization of co-m modules. In section 2, we introduce the S-small and S-essential submodules of a unitary $R$-module $M$ over a commutative ring $R$ with $1\neq 0$ such that S is a multiplicatively closed subset of $R$. We prove that if $M$ is an S-co-m module satisfying the S-DAC and $N\leq M$, then $N\leq ^{S}_{e}M$ if and only if there exists $I\ll ^{S}R$ such that $s(0:_{M}I)\leq N\leq (0 :_{M}I)$ for some $s\in S$. Let $M$ be a faithful S-strong co-m $R$-module. We prove that if $N\ll ^{S}M$ then there exists an ideal $I\leq ^{S…
This paper studies the essential normality of Bergman modules over the intersection of complex ellipsoids, as well as their quotients by monomial ideals.
Let $R$ be a commutative ring and $I\subset R$ a finitely generated ideal. We discuss two definitions of derived $I$-adically complete (also derived $I$-torsion) complexes of $R$-modules which appear in the literature: the idealistic and the sequential one. The two definitions are known to be equivalent for a weakly proregular ideal $I$; we show that they are different otherwise. We argue that the sequential approach works well, but the idealistic one needs to be reinterpreted or properly understood. We also consider $I$-adically flat $R$-modules.
