A characterization of modules locally of finite injective dimension

The Frobenius functor and injective modules

We investigate commutative Noetherian rings of prime characteristic such that the Frobenius functor applied to any injective module is again injective. We characterize the class of one-dimensional local rings with this property and show that it includes all one-dimensional F-pure rings.

Gorenstein injective envelopes of Artinian modules

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.

Red-injective modules

Let $\text{Red}(M)$ be the sum of all reduced submodules of a module $M$. For modules over commutative rings, $\text{Soc}(M)\subseteq \text{Red}(M)$. By drawing motivation from how $\text{Soc}$-injective modules were defined by Amin et. al. in \cite{amin2005}, we introduce $\text{Red}$-injective modules, study their properties and use them to characterize quasi-Frobenius rings and $V$-rings.

Some Remarks on $ϕ$-Dedekind rings and $ϕ$-Prufer rings

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.

On the Gorensetein $(n,d)$-Flat and Gorensetein $(n,d)$-Injective Modules

Let R be a ring. In this paper, Gorenstein (n,d)-flat and (n,d)-Gorenstein (n,d)-injective modules and some of their basic properties are studied. Moreover, some characterizations of rings over Gorenstein (n,d)-flat and (n,d)-Gorenstein (n,d)-injective modules are given. Also, some known results can be obtained as corollaries

The existence of finitely generated modules of finite Gorenstein injective dimension

In this note, we study commutative Noetherian local rings having finitely generated modules of finite Gorenstein injective dimension. In particular, we consider whether such rings are Cohen-Macaulay.

Torsion of injective modules and weakly pro-regular sequences

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…

Weak dimension of FP-injective modules over chain rings

It is proven that the weak dimension of each FP-injective module over a chain ring which is either Archimedean or not semicoherent is less or equal to 2.

Injective Modules under Faithfully Flat Ring Extensions

Let R be a commutative ring and S be an R-algebra. It is well-known that if N is an injective R-module, then Hom(S,N) is an injective S-module. The converse is not true, not even if R is a commutative noetherian local ring and S is its completion, but it is close: It is a special case of our main theorem that in this setting, an R-module N with Ext^i(S,N)=0 for all i>0 is injective if Hom(S,N) is an injective S-module.

Strongly $ϕ$-flat modules, strongly nonnil-injective modules and their homological dimensions

In this paper, we first introduce and study the notions of strongly $ϕ$-flat modules and strongly nonnil-injective modules. And then, we investigate the homology dimensions of modules and rings in terms of these two notions. Finally we give a new homological characterizations of $ϕ$-Dedekind rings and $ϕ$-\Prufer\ rings.

Some criteria of cyclically pure injective modules

The structure of cyclically pure injective modules over a commutative ring $R$ is investigated and several characterizations for them are presented. In particular, we prove that a module $D$ is cyclically pure injective if and only if $D$ is isomorphic to a direct summand of a module of the form $\Hom_R(L,E)$ where $L$ is the direct sum of a family of finitely presented cyclic modules and $E$ is an injective module. Also, we prove that over a quasi-complete Noetherian ring $(R,\fm)$ an $R$-module $D$ is cyclically pure injective if and only if there is a family $\{C_λ\}_{λ\in Λ}$ of cocyclic m…

On Ding injective, Ding projective, and Ding flat modules and complexes

We characterize Ding modules and complexes over Ding-Chen rings. We show that over a Ding-Chen ring R, the Ding projective (resp. Ding injective, resp. Ding flat) R-modules coincide with the Gorenstein projective (resp. Gorenstein injective, resp. Gorenstein flat) modules, which in turn are nothing more than modules appearing as a cycle of an exact complex of projective (resp. injective, resp. flat) modules. We prove a similar characterization for chain complexes of R-modules: A complex is Ding projective (resp. Ding injective, resp. Ding flat) if and only if each component is Ding projective …

On injective dimension of F-finite F-modules and holonomic D-modules

We investigate injective dimension of $F$-finite $F$-modules in characteristic $p$ and holonomic $D$-modules in characteristic 0. One of our main results is the following. If, either $R$ is a regular ring of finite type over an infinite field of characteristic $p>0$ and $M$ is an $F_R$-finite $F_R$-module, or $R=k[x_1,\dots,x_n]$ where $k$ is a field of characteristic 0 and $M$ is a holonomic $D(R,k)$-module, then the injective dimension of $M$ is the same as the dimension of its support.

均匀S伪注入模块

Uniformly S-pseudo-injective modules

本文研究了均匀S伪注入模块的性质，证明了u-S准注入与u-S伪注入之间的等价关系，并引入了与QI环相关的两类环。

Two characterizations of pure injective modules

Let $R$ be a commutative ring with identity and $D$ an $R$-module. It is shown that if $D$ is pure injective, then $D$ is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if $R$ is Noetherian, then $D$ is pure injective if and only if $D$ is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that $D$ is pure injective if and only if there is a family $\{T_λ\}_{λ\in Λ}$ of $R$-algebras which are finitely presented as $R$-modules, such that $D$ is isomorphic to a d…

Gorenstein内射模的正向极限

Direct limits of Gorenstein injective modules

本文研究Gorenstein内射模类对任意正向极限封闭的条件，证明其与左noetherian性、特征模Gorenstein平坦性、覆盖性及纯子模或纯商封闭性等价。

关于广义内射模局部化的一个注记

A note on the localization of generalized injective modules

本文研究 $S$-Noetherian 与 $u$-$S$-Noetherian 环中广义内射模的局部化，证明 $u$-$S$-绝对纯性质在 $S$-凝聚环局部化下保持，并通过例子和反例澄清概念差异、否定相关问题。

A Bass equality for Gorenstein injective dimension of modules finite over homomorphisms

Let $R \to S$ be a local ring homomorphism and $N$ a finitely generated $S$-module. We prove that if the Gorenstein injective dimension of $N$ over $R$ is finite, then it equals the depth of $R$.

S-FP-内射模

S-FP-injective modules

本文在交换环的乘法子集S上引入S-FP-内射模概念，并证明在特定条件下S-诺特性与S-FP-内射模等于S-内射模等价，同时给出S-凝聚环的相应刻画。

On the injective dimension of F-finite modules and holonomic D-modules

Let $R$ be a regular local ring containing a field $k$ of characteristic $p$ and $M$ be an $\mathscr{F}$-finite module. In this paper, we study the injective dimension of $M$. We prove that $\operatorname{dim}_R(M) -1 \leq\operatorname{inj.dim}_R(M)$. If $R = k[[x_1,\ldots,x_n]]$ where $k$ is a field of characteristic $0$ we prove the analogous result for a class of holonomic $\mathscr{D}$-modules which contains local cohomology modules.

Gorenstein内射模类是覆盖类当且仅当它对直极限封闭

The class of Gorenstein injective modules is covering if and only if it is closed under direct limits

该研究证明Gorenstein内射模类的特殊预覆盖性、覆盖性与直极限封闭性三者等价，为Enochs猜想提供新例证，并刻画了使该模类为覆盖类的环的等价条件。

Indecomposable injective modules of finite Malcev rank over local commutative rings

It is proven that each indecomposable injective module over a valuation domain $R$ is polyserial if and only if each maximal immediate extension $\widehat{R}$ of $R$ is of finite rank over the completion $\widetilde{R}$ of $R$ in the $R$-topology. In this case, for each indecomposable injective module $E$, the following invariants are finite and equal: its Malcev rank, its Fleischer rank and its dual Goldie dimension. Similar results are obtained for chain rings satisfying some additional properties. It is also shown that each indecomposable injective module over one Krull-dimensional local No…

论$S$-内射模

On $S$-injective modules

本文在含单位元交换环的乘法子集$S$下提出$S$-内射模概念，给出内射模Baer刻画、平坦模Lambek刻画的$S$-版本，并在特定条件下得到Noether环两类经典刻画的$S$-对应结果。

有限内射维数模的注记

Notes on modules of finite injective dimension

受Bass猜想的启发，研究有限生成且具有有限内射维数的模及其对环的额外约束，包括约化性、正规性、整环性、完全交或Gorenstein等性质，并探讨模的自反性、无挠性、张量积和内射维数等。

Gorenstein injective filtrations over Cohen-Macaulay rings with dualizing modules

Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module, Gorenstein injective modules admit similar filtrations. We also investigate Tor-modules of Gorenstein injective modules over such rings. This extends work of Enochs and Huang over Gorenstein rings. Furthermore, we give examples showing the following: (1) the class of Gorenstein injective $R$-modules need not be closed under tensor products, even when $R$ is local a…

A dichotomy for the injective dimension of $F$-finite $F$-modules and holonomic $D$-modules

Let $M$ be either an $F$-finite $F$-module over a noetherian regular ring of characteristic $p > 0$ or a holonomic $D$-module over a formal power series ring over a field of characteristic zero. We prove that $\injdim_R M$ enjoys a dichotomy property: it has only two possible values, $\dim \Supp_R M - 1$ or $\dim \Supp_R M$.

Rings with finite n-Weak injective dimension and (n,k)-Weak cotorsion modules

Let R be a ring and n,k be two non-negative integers. As an extension of several known notions, we introduce and study (n,k)-weak cotorsion modules using the class of right R-modules with n-weak flat dimensions at most k. Various examples and applications are also given.

局部环上模具有有限内射维数的判据

Criteria for finite injective dimension of modules over a local ring

本文在交换诺特局部环上，利用满足重数与型不等式的 Cohen--Macaulay 模及有限多个 Ext 模的消失来刻画模的有限内射维数，并由此恢复已有结果、给出新的充分条件。

到/自自由模与内射模的不可约同态的结构

Structure of irreducible homomorphisms to/from free and injective modules

本文研究交换诺特局部环上自由模与内射模的不可约同态结构，扩展了已有工作，并探讨完备情形下阿廷模范畴中的不可约单态射与满态射。

Gorenstein injective precovers, covers, and envelopes

We give a sufficient condition for the class of Gorenstein injective modules be precovering: if $R$ is right noetherian and if the class of Gorenstein injective modules, $\mathcal{GI}$, is closed under filtrations, then $\mathcal{GI}$ is precovering in $R-Mod$. The converse is also true when we assume that $\mathcal{GI}$ is covering. We extend our results to the category of complexes. We prove that if the class of Gorenstein injective modules is closed under filtrations then the class of Gorenstein injective complexes is precovering in $Ch(R)$. We also give a sufficient condition for the exist…

Injective modules and torsion functors

A commutative ring is said to have ITI with respect to an ideal a if the a-torsion functor preserves injectivity of modules. Classes of rings with ITI or without ITI with respect to certain sets of ideals are identified. Behaviour of ITI under formation of rings of fractions, tensor products and idealisation is studied. Applications to local cohomology over non-noetherian rings are given.

Module structure of an injective resolution

Let A be the ring obtained by localizing the polynomial ring k[X,Y,Z,W] over a field k at the maximal ideal (X,Y,Z,W) and modulo the ideal (XW-YZ). Let p be the ideal of A generated by X and Y. We study the module structure of a minimal injective resolution of A/p in details using local cohomology. Applications include the description of Ext^i(M,A/p), where M is a module constructed by Dutta, Hochster and McLaughlin, and the Yoneda product of Ext^*(A/p,A/p).

On injective resolutions of local cohomology modules

Let $K$ be a field of characteristic zero and let $R = K[X_1,\ldots,X_n]$. Let $I$ be an ideal in $R$ and let $M = H^i_I(R)$ be the $i^{th}$-local cohomology module of $R$ with respect to $I$. Let $c = \ injdim \ M$. We prove that if $P$ is a prime ideal in $R$ with Bass number $μ_c(P,M) > 0$ then $P$ is a maximal ideal in $R$.

拟内射维数

Quasi-injective dimension

本文引入拟内射维数以推广内射维数，并证明其在局部环上关于深度、Cohen-Macaulay、Gorenstein 性质及与拟投射维数等价性的多个经典结论。

A Note on the injective dimension of local cohomology modules

For a noetherian ring R we call an R-module M cofinite if there exists an ideal I of R such that M is I-cofinite; we show that every cofinite module M satisfies dim_R(M)<=injdimR(M). As an application we study the question which local cohomology modules H^i_I(R) satisfy injdim_R(H^i_I(R)) = dim_R(H^i_I(R)). There are two situations where the answer is positive. On the other hand we present two counter-examples, the failure in these two examples coming from different reasons.

Local cohomology modules and Gorenstein injectivity with respect to a semidualizing module

Let $(R,\fm)$ be a local ring and let $C$ be a semidualizing $R$--module. In this paper, we are concerned in $C$--injective and $G_{C}$--injective dimensions of certain local cohomology modules of $R$. Firstly, the injective dimension of $C$ and the above quantities of dimensions is compared. Then, as an application of the above comparisons, a characterization of a dualizing module of $R$ is given. Finally, it is shown that if $R$ is Cohen-Macaulay of dimension $d$ such that $\H_{\fm}^{d}(C)$ is $C$--injective, then $R$ is Gorernstein. This is an answer to the question which was recently prese…

On injective and Gorenstein injective dimensions of local cohomology modules

Let $(R,\fm)$ be a commutative Noetherian local ring and let $M$ be an $R$-module which is a relative Cohen-Macaulay with respect to a proper ideal $\fa$ of $R$ and set $n:=\h_{M}\fa$. We prove that $\ind M<\infty$ if and only if $\ind\H^{n}_\fa(M)<\infty$ and that $\ind\H^{n}_\fa(M)=\ind M-n$. We also prove that if $R$ has a dualizing complex and $\Gid_{R} M<\infty$, then $\Gid_{R}\H^{n}_\fa(M)<\infty$ and $\Gid_{R}\H^{n}_\fa(M)=\Gid_{R} M-n$. Moreover if $R$ and $M$ are Cohen-Macaulay, then it is proved that $\Gid_{R} M<\infty$ whenever $\Gid_{R}\H^{n}_\fa(M)<\infty$. Next, for a finitely ge…

Tests for injectivity of modules over commutative rings

It is proved that a module M over a commutative noetherian ring R is injective if Ext^i((R/p)_p,M)=0 holds for every i\ge 1 and every prime ideal p in R. This leads to the following characterization of injective modules: If F is faithfully flat, then a module M such that Hom(F,M) is injective and Ext^i(F,M)=0 for all i\ge 1 is injective. A limited version of this characterization is also proved for certain non-noetherian rings.

On $L_{n}$-Injective Modules and $L_{n}$-Injective Dimensions

Let $R$ be a ring, and $n$ a fixed nonnegative integer. An $R$-module $W$ is called $L_{n}$-injective if ${\rm Ext}_{R}^{1}(M,W)=0$ for any $R$-module $M$ with flat dimension at most $n$. In this paper, we prove first that ($\mathcal{F}_{n},\mathcal{L}_{n}$) is a complete hereditary cotorsion theory, where $\mathcal{F}_n$ (resp. $\mathcal{L}_n$) denotes the class of all $R$-modules with flat dimension at most $n$ (resp. $L_{n}$-injective $R$-modules). Then we introduce the $L_{n}$-injective dimension of a module and $L_n$-global dimension of a ring. Finally, over rings with weak global dimensi…

本质上同调模

Essential cohomology modules

本文通过 e-正合序列推广内射模为 e-内射模，并借助其构造本质上同调模，证明挠函子在无挠模上 e-正合，并揭示 e-上同调与经典上同调在消失性上的差异。

The Existence of Relative pure Injective Envelopes

Let $\mathcal{S}$ be a class of finitely presented $R$-modules such that $R\in \mathcal{S}$ and $\mathcal{S}$ has a subset $\mathcal{S}^*,$ with the property that for any $U\in \mathcal{S}$ there is a $U^*\in \mathcal{S}^*$ with $U^*\cong U.$ We show that the class of $\mathcal{S}$-pure injective $R$-modules is preenveloping. As an application, we deduce that the left global $\mathcal{S}$-pure projective dimension of $R$ is equal to its left global $\mathcal{S}$-pure injective dimension. As our main result, we prove that, in fact, the class of $\mathcal{S}$-pure injective $R$-modules is envelo…

收缩自同态下的正则性、局部上同调与内射模

Regularity, local cohomology and injective modules under contracting endomorphisms

研究将特征\(p\)环的结果扩展到含收缩自同态的环，利用\(\text{Hom}_R(^{\phi}R,-)\)得到正则环新特征，还研究\((-) \otimes_R {^{\phi}}R\)在局部上同调模与内射模上的行为。

弱Ding内射复形

Weakly Ding injective complexes

在左凝聚环上定义弱Ding内射复形类，研究其生成的余挠对，并展示为阿贝尔模型结构的一部分；在Ding-Chen环上证明稳定链复形范畴紧生成且三角等价于四个Frobenius范畴的稳定范畴。

Embedding modules of finite homological dimension

This paper builds on work of Hochster and Yao that provides nice embeddings for finitely generated modules of finite G-dimension, finite projective dimension, or locally finite injective dimension. We extend these results by providing similar embeddings in the relative setting, that is, for certain modules of finite G_C-dimension, finite P_C-projective dimension, locally finite GI_C-injective dimension, or locally finite I_C-injective dimension where C is a semidualizing module. Along the way, we extend some results for modules of finite homological dimension to modules of locally finite homol…

Gorenstein injective envelopes and covers over two sided noetherian rings

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.

FP-injectivity of factors of injective modules

It is shown that a ring is left semihereditary if and only each homomorphic image of its injective hull as left module is FP-injective. It is also proven that a commutative ring R is reduced and arithmetical if and only if E/U if FP-injective for any FP-injective R-module E and for any submodule U of finite Goldie dimension. A characterization of commutative rings for which each module of finite Goldie dimension is of injective dimension at most one is given. Let R be a chain ring and Z its subset of zerodivisors. It is proven that E/U is FP-injective for each FP-injective R-module E and each …

Existence of covers and envelopes of a left orthogonal class and its right orthogonal class of modules

In this paper, we investigate the notions of $\mathcal{X}^\bot$-projective, $\mathcal{X}$-injective and $\mathcal{X}$-flat modules and give some characterizations of these modules, where $\mathcal{X}$ is a class of left $R$-modules. We prove that the class of all $\mathcal{X}^\bot$-projective modules is Kaplansky. Further, if the class of all $\mathcal{X}$-projective $R$-modules is closed under direct limits, we show the existence of $\mathcal{X}^\bot$-projective covers and $\mathcal{X}$-injective envelopes over a $\mathcal{X}^\bot$-hereditary ring $R.$ Moreover, we decompose a $\mathcal{X}^\b…

Stable Local Cohomology

Let $R$ be a Gorenstein local ring, $\frak{a}$ an ideal in $R$, and $M$ an $R$-module. The local cohomology of $M$ supported at $\frak{a}$ can be computed by applying the $\frak{a}$-torsion functor to an injective resolution of $M$. Since $R$ is Gorenstein, $M$ has a complete injective resolution, so it is natural to ask what one gets by applying the $\frak{a}$-torsion functor to it. Following this lead, we define stable local cohomology for modules with complete injective resolutions. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this b…

Canonical Filtrations of Gorenstein Injective Modules

The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So perhaps it is surprising that tensor products of Gorenstein injective modules over Gorenstein rings of finite Krull dimension are Gorenstein injective. Our main result is in support of the princ…

Gorenstein平坦预包络与弱Ding内射覆盖

Gorenstein flat preenvelopes and weakly Ding injective covers

本文在凝聚环框架下研究Gorenstein平坦模的预包络存在性，引入弱Ding内射模类，证明三个等价命题并给出GF为预包络类的充分条件，拓展了Ding-Chen环上的相关结果。

Category of n-weak injective and n-weak flat modules with respect to special super presented modules

Let $R$ be a ring and $n$, $k$ two non-negative integers. In this paper, we introduce the concepts of $n$-weak injective and $n$-weak flat modules and via the notion of special super finitely presented modules, we obtain some characterizations of these modules. We also investigate two classes of modules with richer contents, namely $\mathcal{WI}_k^n(R)$ and $\mathcal{WF}_k^n(R^{op})$ which are larger than that of modules with weak injective and weak flat dimensions less than or equal to $k$. Then on any arbitrary ring, we study the existence of $\mathcal{WI}_k^n(R)$ and $\mathcal{WF}_k^n(R^{op…

持久模的平坦覆盖与内射包

Flat covers and injective hulls of persistence modules

本文利用 Enochs–Xu 的平坦余挠模理论，在持久模的内射包与平坦覆盖及其极小分解间建立 Matlis 对偶，从而可处理非驯顺或非逐点有限维情形。

伪Krull序上的内射模

Injective modules over pseudo-krullian orders

本文引入伪Krull序，研究其模范畴关于伪同构的Serre商，描述了商范畴中的内射对象及全局同调维数，推广了I. Beck关于Krull环的结论，并确立了特定条件下极大Cohen-Macaulay模范畴的全局同调维数。

关于内射模逆向极限的一个注记

A note on inverse limits of injective modules

本文证明任意模可表示为内射模的逆向极限（分别以 $\omega_1$ 和 $\omega$ 为指标），回答 Bergman 的两个问题，并刻画了相应直和项类在交换 Noether 环上恰为 Enochs 余挠模的条件。