Related papers: On Regular Fusible Modules

In this note we introduce and study basic properties of two types of modules over a commutative noetherian ring $R$ of positive prime characteristic. The first is the category of modules of finite $F$-type. These objects include reflexive…

Let $FI$ be a skeleton of the category of finite sets and injective maps, and $FI^m$ the product of $m$ copies of $FI$. We prove that if an $FI^m$-module is generated in degree $\leqslant d$ and related in degree $\leqslant r$, then its…

Let $T_R(M)$ be a tensor ring and $\mathcal{X}$, $\mathcal{Y}$ be two classes of $R$-modules. Under certain conditions, we prove that a $T_R(M)$-module $(A, u)$ is $Ind(\mathcal{X})$-Gorenstein projective if and only if $u$ is monomorphic…

$\textbf{Theorem 1.2.}$ For a ring $A$, the following conditions are equivalent. $\textbf{1)}$ $A$ is a right automorphism-invariant right non-singular ring. $\textbf{2)}$ $A$ is a right automorphism-invariant regular ring. $\textbf{3)}$…

Let $R$ be a valuation ring and let $Q$ be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if $Q$ is maximal (respectively artinian). It is shown that each…

Let $R$ be a commutative ring. We investigate $R$-modules which can be written as \emph{finite} sums of {\it {second}} $R$-submodules (we call them \emph{second representable}). We provide sufficient conditions for an $R$-module $M$ to be…

Over a commutative noetherian ring $R$ of finite Krull dimension, we show that every complex of flat cotorsion $R$-modules decomposes as a direct sum of a minimal complex and a contractible complex. Moreover, we define the notion of a…

The class of radial fuzzy systems is introduced. The fuzzy systems in this class use radial functions to implement membership functions of fuzzy sets and exhibit a shape preservation property in antecedents of their rules. The property is…

Let $R$ be a commutative noetherian ring and $f_{1}, ..., f_{r} \in R$. In this article we give (cf. the Theorem in \S2) a criterion for $f_{1}, ..., f_{r}$ to be regular sequence for a finitely generated module over $R$ which strengthens…

Let $R$ be a ring. It is proved that an $R$-module $M$ is Baer (resp. dual Baer) if and only if every exact sequence $0\rightarrow X\rightarrow M\rightarrow Y\rightarrow 0$ with $Y\in$ Cog$(M_R)$ (resp. $X\in$ Gen$(M_R)$) splits. This shows…

The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is…

Let R be a commutative ring with identity and M be an R-module. The purpose of this paper is to introduced the dual notion of z-submodules of M and some of extensions. Moreover, we investigate some properties of these classes of modules…

Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S =$ End$_R(M)$. In this paper we introduce dual $\pi$-Rickart modules as a generalization of $\pi$-regular rings as well as that of dual Rickart modules. The…

Let $M$ be an $R$-module and $c$ the function from $M$ to the ideals of $R$ defined by $c(x) = \cap \lbrace I \colon I \text{is an ideal of} R \text{and} x \in IM \rbrace $. $M$ is said to be a content $R$-module if $x \in c(x)M $, for all…

The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition "$(\sqrt[p]{{\cal{P}}M}:M)={\cal{P}}$" for…

Based upon properties of ordinal length, we introduce a new class of modules, the binary modules, and study their endomorphism ring. The nilpotent endomorphisms form a two-sided ideal, and after factoring this out, we get a commutative…

We introduce the notion of a regular action in the category of conformal modules over Lie conformal algebras with Virasoro elements. We show that a finite conformal module over the general conformal algebra $\mathfrak{gc}_1$ (resp.,…

Let $T_R(M)$ be a tensor ring, where $R$ is a ring and $M$ is an $N$-nilpotent $R$-bimodule. Under certain conditions, we characterize the Gorenstein flat-cotorsion modules over $T_R(M)$, showing that a $T_R(M)$-module $(X, u)$ is…

For a commutative ring $S$ and self-orthogonal subcategory $\mathsf{C}$ of $\mathsf{Mod}(S)$, we consider matrix factorizations whose modules belong to $\mathsf{C}$. Let $f\in S$ be a regular element. If $f$ is $M$-regular for every $M\in…

Let $R$ be a ring and $S$ a multiplicative subset of $R$. We introduce and study the notions of ($u$-)$S$-$w$-Noetherian modules and ($u$-)$S$-$w$-principal ideal modules. Some characterizations of these new concepts are given.