Let $T$ be a Noetherian ring and $f$ a nonzerodivisor on $T$. We study concrete necessary and sufficient conditions for a module over $R=T/(f)$ to be weakly liftable to $T$, in the sense of Auslander, Ding and Solberg. We focus on cyclic modules and get various positive and negative results on the lifting and weak lifting problems. For a module over $T$ we define the loci for certain properties: liftable, weakly liftable, having finite projective dimension and study their relationships.
Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra over a commutative ring R. Assume that (a) B=A[X_1,...,X_n] is a polynomial extension of A, where X_1,...,X_n are variables of positive degrees; or (b) A is a divided power DG R-algebra and B=A<X_1,...,X_n> is a free extension of A obtained by adjunction of variables X_1,...,X_n of positive degrees. In this paper, we study naive liftability of DG modules along the natural injection A-->B using the notions of diagonal ideals and homotopy limits. We prove that if N is a bounded below semifree DG B-m…
Let $B = A< X | dX=t >$ be an extended DG algebra by the adjunction of variable of positive even degree $n$, and let $N$ be a semi-free DG $B$-module that is assumed to be bounded below as a graded module. We prove in this paper that $N$ is liftable to $A$ if $Ext_B^{n+1}(N,N)=0$. Furthermore such a lifting is unique up to DG isomorphisms if $Ext_B^{n}(N,N)=0$.
The notion of weakly Laskerian modules was introduced recently by the authors. Let $R$ be a commutative Noetherian ring with identity, $\fa$ an ideal of $R$, and $M$ a weakly Laskerian module. It is shown that if $\fa$ is principal, then the set of associated primes of the local cohomology module $H_{\fa}^i(M)$ is finite for all $i\geq 0$. We also prove that when $R$ is local, then $\Ass_R(H_{\fa}^i(M))$ is finite for all $i\geq 0$ in the following cases: (1) $\dim R\leq 3$, (2) $\dim R/\fa\leq 1$, (3) $M$ is Cohen-Macaulay and for any ideal $\fb$, with $l=\grade(\fb,M)$, $\Hom_R(R/\fb,H_{\fb}…
Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. By $J(R),$ we denote the Jacobson radical of $R$. The purpose of this paper is to introduce the concept of weakly $J$-submodules generalizing $J$-submodules. We call a proper submodule $N$ of $M$ a weakly $J$-submodule if whenever $0\neq rm\in N$ for $r\in R$ and $m\in M$, then $r\in(J(R)M:M)$ or $m\in N.$ Various properties and characterizations of weakly $J$-submodules are investigated especially in the case of multiplication modules.
Let $R$ be a commutative Noetherian ring, $Φ$ a system of ideals of $R$ and $I\in Φ$. Let $M$ be an $R$-module (not necessary $I$-torsion) such that $\dim M\leq 1$, then the $R$-module $\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for all $i\geq 0$, if and only if the $R$-module $\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for $i=0, 1$. Let $t\in\Bbb{N}_0$ be an integer and $M$ an $R$-module such that $\Ext^i_R(R/I,M)$ is weakly Laskerian for all $i\leq t+1$. We prove that if the $R$-module $\lc^{i}_Φ(M)$ is ${\rm FD_{\leq 1}}$ for all $i<t$, then $\lc^{i}_Φ(M)$ is $Φ$-weakly cofinite for all $i<t$ a…
Let $R$ be a commutative Noetherian ring, $\fa$ be an ideal of $R$ and $M$ be an $R$-module. The main purpose of this paper is to answer the Hartshorn's questions in the class of weakly Laskerian modules. It is shown that if $s\geq 1$ is a positive integer such that $\Ext^j_R(R/\fa, M)$ is weakly Laskerian for all $j\leq s$ and the $R$-module $H^i_\fa(M)$ is $FD_{\leq 1}$ for all $i < s$, then the $R$-module $H^i_\fa(M)$ is $\fa$-weakly cofinite for all $i <s$. In addition, we show that the category of all $\fa$-weakly cofinite $FD_{\leq 1}$ $R$-modules is an Abelian subcategory of the categor…
The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal $\frak a$ of a Noetherian ring $R$. It is shown that an $R$-module $M$ is cofinite with respect to $\frak a$, if and only if, $\Ext^i_R(R/\frak a,M)$ is finitely generated for all $i\leq {\rm cd}(\frak a,M)+1$, whenever $\dim R/\frak a=1$. In addition, we show that if $M$ is finitely generated and $H^i_{\frak a}(M)$ are weakly Laskerian for all $i\leq t-1$, then $H^i_{\frak a}(M)$ are ${\frak a}$-cofinite for all $i\leq t-1$ and for any minimax submodule $K$ of $H^{t}_{\fr…
Let $R$ a commutative ring, $\mathfrak{a} \subset R$ an ideal, $I$ an injective $R$-module and $S \subset R$ a multiplicatively closed set. When $R$ is Noetherian it is well-known that the $\mathfrak{a}$-torsion sub-module $Γ_{\mathfrak{a}}(I)$, the factor module $I/Γ_{\mathfrak{a}}(I)$ and the localization $I_S$ are again injective $R$-modules. We investigate these properties in the case of a commutative ring $R$ by means of a notion of relatively-$\mathfrak{a}$-injective $R$-modules. In particular we get another characterization of weakly pro-regular sequences in terms of relatively injectiv…
Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=ϕ$ to be weakly $S$-prime if there exists $s\in S$ such that whenever $a\in R$ and $m\in M$ with $0\neq am\in N$, then either $sa\in(N:_{R}M)$ or $sm\in N$. Many properties, examples and characterizations of weakly $S$-prime submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotie…
In this study, all rings are commutative with non-zero identity and all modules are considered to be unital. Let $M$ be a left $R$-module. A proper submodule $N$ of $M$ is called an $S$-$weakly$ $prime$ submodule if $0_{M}\neq f(m)\in N$ implies that either $m\in N$ or $f(M)\subseteq N,$ where $f\in S=End(M)$ and $m\in M$. Some results concerning $S$-prime and $S$-weakly prime submodules are obtained. Then we study $S$-prime and $S$-weakly prime submodules of multiplication modules. Also for $R$-modules $M_{1}$ and $M_{2},$ we examine $S$-prime and $S$-weakly prime submodules of $M=M_{1}\times…
Let $I$ be an ideal of a Noetherian ring R and M be a finitely generated R-module. We introduce the class of extension modules of finitely generated modules by the class of all modules $T$ with $\dim T\leq n$ and we show it by ${\rm FD_{\leq n}}$ where $n\geq -1$ is an integer. We prove that for any ${\rm FD_{\leq 0}}$(or minimax) submodule N of $H^t_I(M)$ the R-modules ${\rm Hom}_R(R/I,H^{t}_I(M)/N) {\rm and} {\rm Ext}^1_R(R/I,H^{t}_I(M)/N)$ are finitely generated, whenever the modules $H^0_I(M)$, $H^1_I(M)$, ..., $H^{t-1}_I(M)$ are ${\rm FD_{\leq 1}}$ (or weakly Laskerian). As a consequence,…
We prove that if a local ring admits a (pd-)test module of finite complete intersection dimension, then it is a complete intersection ring. This answers, positively, a question proposed by Celikbas, Dao and Takahashi. To this aim, we first investigate another question raised by Celikbas and Sather- Wagstaff concerning ascent properties of (pd-)test complexes under weakly regular homomorphisms.
Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ to be weakly $S$-primary if there exists $s\in S$ such that whenever $a\in R$ and $m\in M$ with $0\neq am\in N$, then either $sa\in\sqrt{(N:_{R}M)}$ or $sm\in N$. We present various properties and characterizations of this concept (especially in finitely generated faithful multiplication modules). Moreover, the behavior of this structure under module homomorphisms, localizations, q…
Let $R$ be a commutative noetherian ring and let $\frak a$ be an ideal of $R$. In this paper, we study a certain condition, namely $C_{\frak a}$, introduced by Aghapournahr and Melkersson, on the extension of two subcategories of $R$-modules. We extend and generalize some of the main results of Yoshizawa [Y1,Y2]. As an example of extension of subcategories, we study the weakly Laskerian modules and we find some conditions under which the local cohomology modules of a weakly Laskerian module lie in an arbitrary Serre subcategory. Eventually, we investigate the cofiniteness of the local cohomolo…
Let $R$ be a $G$-graded ring. In this article, we introduce two new concepts on graded rings, namely, weakly graded rings and invertible graded rings, and we discuss the relations between these concepts and several properties of graded rings. Also, we study the concept of weakly crossed products and study some properties defined on weakly crossed product and give the relationship between this new concept and several properties of graded rings. Moreover, in this article, we give a generalization for the concept of graded essential submodules, and introduce the concept of graded semi-uniform mod…
Let $R$ denote a commutative Noetherian (not necessarily local) ring, $M$ an arbitrary $R$-module and $I$ an ideal of $R$ of dimension one. It is shown that the $R$-module $\Ext^i_R(R/I,M)$ is finitely generated (resp. weakly Laskerian) for all $i\leq {\rm cd}(I,M)+1$ if and only if the local cohomology module $H^i_I(M)$ is $I$-cofinite (resp. $I$-weakly cofinite) for all $i$. Also, we show that when $I$ is an arbitrary ideal and $M$ is finitely generated module such that the $R$-module $H^i_I(M)$ is weakly Laskerian for all $i\leq t-1$, then $H^i_I(M)$ is $I$-cofinite for all $i\leq t-1$ and …
The existence of Ulrich modules for local domains has been a difficult and elusive open question. For over thirty years, it was unknown whether local domains always have Ulrich modules. In this paper, we answer the question of existence for both Ulrich modules and weakly lim Ulrich sequences -- a weaker notion recently introduced by Ma -- in the negative. We construct many local domains in all dimensions $d \geq 2$ that do not have any Ulrich modules. Moreover, we show that when $d = 2$, these local domains do not have weakly lim Ulrich sequences. A key insight in our proofs is the classificat…
Let $N$ be a submodule of a finitely generated module $M$ over a Noetherian ring. A method for the computation of the submodule generated by the envelope of $N$ is given. The relations between weakly prime submodules and their envelopes are investigated. Using these relations, a description of the weakly radical of a submodule is obtained. The results are illustrated by examples.
Let R be a commutative ring with identity. We investigate some ring-theoretic properties of weakly Laskerian R-modules. Our results indicate that weakly Laskerian rings behave as Noetherian ones in many respects. However, we provide some examples to illustrate the strange behavior of these rings in some other respects.
The aim of this paper is to describe the classes of strongly flat and weakly cotorsion modules with respect to a multiplicative subset or a finite collection of multiplicative subsets in a commutative ring. The strongly flat modules are characterized by a set of conditions, while the weakly cotorsion modules are produced by a generation procedure. Passing to the collection of all countable multiplicative subsets, we define quite flat and almost cotorsion modules, and show that, over a Noetherian ring with countable spectrum, all flat modules are quite flat and all almost cotorsion modules are …
We introduce the notion of Burch submodules and weakly $\mathfrak m$-full submodules of modules over local rings and study their properties. One of our main results shows that Burch submodules satisfy 2-Tor rigid and test property. We also show that over a local ring $(R,\mathfrak m)$ a submodule $M$ of a finitely generated $R$-module $X$, such that either $M=\mathfrak m X$ or $M(\subseteq \mathfrak m X$) is weakly $\mathfrak m$-full in $X$, is 1-Tor rigid and a test module provided that $X$ is faithful (and $X/M$ has finite length when $M$ is weakly $\mathfrak m$-full). As an application, we …
Let A be a noetherian AS regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and grA the category of finitely generated graded left A-modules. Following Jorgensen, we define the Castelnuovo-Mumford regularity reg(X) of a complex X \in D^b(grA) in terms of the local cohomologies or the minimal projective resolution of X. Let A^! be the quadratic dual ring of A. For the Koszul duality functor DG : D^b(grA) -> D^b(grA^!), we have reg(X) = max {i | H^i(DG (X)) \ne 0}. Using these concepts, we study weakly Koszul modules (= componentwise linear modules) over A^…
Let $R$ be a commutative ring, $M$ an $R$-module and $φ_a$ be the endomorphism of $M$ given by right multiplication by $a\in R$. We say that $M$ is {\it weakly-morphic} if $M/φ_a(M)\cong \ker(φ_a)$ as $R$-modules for every $a$. We study these modules and use them to characterise the rings $R/\text{Ann}_R(M)$, where $\text{Ann}_R(M)$ is the right annihilator of $M$. A kernel-direct or image-direct module $M$ is weakly-morphic if and only if each element of $R/\text{Ann}_R(M)$ is regular as an endomorphism element of $M$. If $M$ is a weakly-morphic module over an integral domain $R$, then $M$ is…
In this paper, all rings are commutative with nonzero identity. Let $M$ be an $R$-module. A proper submodule $N$ of $M$ is called a classical prime submodule, if for each $m \in M$ and elements $a,b\in R$, $abm\in N$ implies that $am\in N$ or $bm\in N$. We introduce the concept of "weakly classical prime submodules." A proper submodule $N$ of $M$ is a weakly classical prime submodule if whenever $a,b\in R$ and $m\in M$ with $0\neq abm\in N$, then $am\in N$ or $bm\in N$.
The weakly Arf $(S_2)$-ification of a commutative Noetherian ring $R$ is considered to be a birational extension which is good next to the normalization. The weakly Arf property (WAP for short) of $R$ was introduced in 1971 by J. Lipman with his famous paper [12], and recently rediscovered by [4], being closely explored with further developments. The present paper aims at constructing, for a given Noetherian ring $R$ which satisfies certain mild conditions, the smallest module-finite birational extension of $R$ which satisfies WAP and the condition $(S_2)$ of Serre. We shall call this extensio…
Let $R$ be a ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. Let $Γ_{\mathfrak{a}}$ denote the $\mathfrak{a}$-torsion functor. Conditions are given for the (weakly) associated primes of $Γ_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ containing $\mathfrak{a}$, and for the (weakly) associated primes of $M/Γ_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ not containing $\mathfrak{a}$.
Let $\mathfrak{a}$ denote an ideal of a commutative Noetherian ring $R$. Let $M$ and $N$ be two $R$-modules. In this paper, we give partial answers on the extension of Hartshorne's conjecture about the cofiniteness of torsion and extension functors. For this purpose, we study the cofiniteness of the generalized local cohomology module $H^{i}_{\mathfrak{a}}(M, N)$ for a new class of modules, called $\mathfrak{a}$-weakly finite modules, in the local and non-local case. Furthermore, we derive some results on attached primes of top generalized local cohomology modules.
Let $(R,\mathfrak m)$ be a Noetherian local ring, $I$ an ideal of $R$ and $M$ a weakly finite or a coatomic $R$-module of dimension $n$. In this article, we resolve the Artinianness and non-Artinianness of top local cohomology modules, $H^{cd(I,M)}_I(M)$, in all cases except in the case $cd(I,M)=n-1$ and $dim(R/{I+Ann(M)})>1$ for which we have some sorter results under certain conditions.
Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations of these distinct notions for modules in terms of both (weakly-)morphic modules and reduced modules. Furthermore, module theoretic settings are established where these in general distinct notions turn out to be indistinguishable.
In this paper, we give a generalization for weakly primary submodules called $I$-primary submodule and we study some properties of it. We give some characterizations of $I$-primary submodules. Also we establish the situation of $I$-primary submodules in module localizations and decomposition modules.
The main aim of this paper is to classify Ulrich ideals and Ulrich modules over two-dimensional Gorenstein rational singularities (rational double points) from a geometric point of view. To achieve this purpose, we introduce the notion of (weakly) special Cohen-Macaulay modules with respect to ideals, and study the relationship between those modules and Ulrich modules with respect to good ideals.
