Related papers: A new characterization of the weakly Laskerian (FS…

Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological…

We prove that if M is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring, and if the ith local cohomology module of M is zero unless i = d, i = 0, and i = r for some r strictly between 0…

We study classes of modules closed under direct sums, $\mathcal{M}$-submodules and $\mathcal{M}$-epimorphic images where $\mathcal{M}$ is either the class of embeddings, $RD$-embeddings or pure embeddings. We show that the…

This paper centers around Artinianness of the local cohomology of $ZD$-modules. Let $\fa$ be an ideal of a commutative Noetherian ring $R$. The notion of $\fa$-relative Goldie dimension of an $R$-module $M$, as a generalization of that of…

Let $R$ be a commutative Noetherian local ring of prime characteristic $p$. The main purposes of this paper are to show that if the injective envelope $E$ of the simple $R$-module has a structure as a torsion-free left module over the…

Let $R$ be a commutative ring with $1\neq0$. In this article, we introduce the concept of weakly $(m,n)-$closed $\delta-$primary ideals of $R$ and explore its basic properties. We show that $I\bowtie^{f}J$ is a weakly $(m,n)-$closed…

Let $K$ be a field, and let $R = K[X]$ be the polynomial ring in an infinite collection $X$ of indeterminates over $K$. Let ${\mathfrak S}_{X}$ be the symmetric group of $X$. The group ${\mathfrak S}_{X}$ acts naturally on $R$, and this in…

Let $P$ be a finitely generated commutative semiring. It was shown recently that if $P$ is a parasemifield (i.e. the multiplicative reduct of $P$ is a group) then $P$ cannot contain the positive rationals $\mathbb{Q}^+$ as its subsemiring.…

For finitely generated modules $N \subsetneq M$ over a Noetherian ring $R$, we study the following properties about primary decomposition: (1) The Compatibility property, which says that if $\ass (M/N)=\{P_1, P_2, ..., P_s\}$ and $Q_i$ is a…

Let $R$ be a (commutative Noetherian) local ring of prime characteristic that is $F$-pure. This paper studies a certain finite set ${\mathcal I}$ of radical ideals of $R$ that is naturally defined by the injective envelope of the simple…

In this paper, we characterize several properties of commutative notherian local rings in terms of the left perpendicular category of the category of finitely generated modules of finite projective dimension. As an application we prove that…

It is well-known that a ring R is semiperfect if and only if R as a left (or as a right) R-module is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e.,…

Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$, $\mathcal{S}$ a Serre subcategory of $R$-modules satisfying the condition $C_\mathfrak{a}$ and $\mathcal{N}$ the subcategory of finitely generated $R$-modules. In this…

Let $R$ be a commutative ring with identity and $M$ a unitary $R$-module. The purpose of this paper is to introduce the concept of semi-$n$-submodules as an extension of semi $n$-ideals and $n$-submodules. A proper submodule $N$ of $M$ is…

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The aim of this paper is to introduce the notion of fully S-idempotent modules as a generalization of fully idempotent modules and investigate some…

Let $R$ be a strong $n$-coherent ring such that each finitely $n$-presented $R$-module has finite projective dimension. We consider $\mathcal{FP}_{n}(R)$ the full subcategory of $R$-Mod of finitely $n$-presented modules. We prove that…

In this paper, we introduce a new concept in Nil-semicommutative modules and present it as an extension of Nil-semicommutative rings to modules. We prove that the class of Nil-semicommutative modules is contained in the class of Weakly…

Let $A$ be a commutative ring with identity. A proper submodule $N$ of $A$-module $M$ is said to be prime submodule if $ax \in N$ where $a \in A, x \in M$, implies $x \in N$ or $aM \subseteq N$. A proper submodule $N \subset M$ is said to…

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ a finitely generated $R$-module with $\dim_R(M)=d$. Denote by $\depth_R(I,M)$ the depth of $M$ in $I$. In \cite{HT}, C. Huneke and V. Trivedi proved that if $R$ is a…

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ an arbitrary $R$-module and $X$ a finite $R$-module. We prove that the category of $\fa$-cominimax modules is a Melkersson subcategory of $R$-modules whenever $\dim R\leq…