Modules Whose Small Submodules Have Krull Dimension

关于弱经典1-吸收素子模

On weakly classical 1-absorbing prime submodules

本文引入并研究弱经典1-吸收素子模概念，给出其例子与性质，并借助张量积和Jacobson根刻画了所有真子模均满足该条件的模类。

纯子模在直和项中本质的模

On modules whose pure submodules are essential in direct summands

本文引入纯扩张模概念，细化经典扩张模，证明其在冯·诺依曼正则环上与扩张模一致，并推广奥索夫斯基-史密斯定理，给出中心拟态性与中心态性一致的充分条件。

Attaching topological spaces to a module (I): Sobriety and spatial frames of submodules

In this paper we study some frames associated to an $R$-module $M$. We define semiprimitive submodules and we prove that they form an spatial frame canonically isomorphic to the topology of $Max(M)$. We characterize the soberness of $Max(M)$ in terms of the point space of that frame. Beside of this, we study the regularity of an spatial frame associated to $M$ given by annihilator conditions.

Strictly invariant submodules

If $M$ is an $R$-module, we study the submodules $K\leq M$ with the property that $K$ is invariant with respect to all monomorphisms $K\rightarrow M$. Such submodules are called \textsl{strictly invariant}. For the case of $% \mathbb{Z}$-modules (i.e. Abelian groups) we prove that in many situations these submodules are invariant with respect to all homomorphisms $% K\rightarrow M$, submodules which were called \textsl{strongly invariant}.

Amalgamation and extensions of summand absorbing modules over a semiring

A submodule $W$ of $V$ is summand absorbing, if $x + y \in W$ implies $x \in W, \; y \in W $ for any $x, y \in V$. Such submodules often appear in modules over (additively) idempotent semirings, particularly in tropical algebra. This paper studies amalgamation and extensions of these submodules, and more generally of upper bound modules.

Modules Whose Classical Prime Submodules Are Intersections of Maximal Submodules

Commutative rings in which every prime ideal is the intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that {\it classical prime} submodules are the intersection of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings $R$ over which all $R$-modules ar…

On the structure of submodule of finitely generated module over PID

Consider $\mathbb{Z}/8\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$, and the subgroup generated by $(2,1)$, it is a subgroup isomorphic to $\mathbb{Z}/4\mathbb{Z}$. If my theorem holds, it must contained in a cyclic group generated by some element of order $8$ which is impossible. The error in 4.2 is because "preimage" do not preserce the image of $[x\mapsto px]$.

The nilpotency of the prime radical of a Goldie module

With the notion of prime submodule defined by F. Raggi et.al. we prove that the intersection of all prime submodules of a Goldie module $M$, is a nilpotent submodule provided that $M$ is retractable and $M^{(Λ)}$-projective for every index set $Λ$. This extends the well known fact that in a left Goldie ring, the prime radical is nilpotent.

S-素右子模与素回避引理的S-版本

S-Prime Right Submodules and an S-Version of Prime Avoidance

本文引入S-素子模概念，研究其性质与等价刻画，证明乘法右模中对应理想为S-素理想当且仅当子模为S-素子模，并给出素回避引理的S-版本及S-诺特模的判定条件。

Bounded Submodules of Modules

Let $m$, $n$ be positive integers such that $m\leq n$. We consider all pairs $(B,A)$ where $B$ is a finite dimensional $T^n$-bounded $k[T]$-module and $A$ is a submodule of $B$ which is $T^m$-bounded. They form the objects of the submodule category $S_m(k[T]/T^n)$ which is a Krull-Schmidt category with Auslander-Reiten sequences. The case $m=n$ deals with submodules of $k[T]/T^n$-modules and has been studied well. In this manuscript we determine the representation type of the categories $S_m(k[T]/T^n)$ also for the cases where $m<n$: It turns out that there are only finitely many indecomposabl…

When an $\mathscr{S}$-closed submodule is a direct summand

It is well known that a direct sum of CLS-modules is not, in general, a CLS-module. It is proved that if $M=M_1\oplus M_2$, where $M_1$ and $M_2$ are CLS-modules such that $M_1$ and $M_2$ are relatively ojective (or $M_1$ is $M_2$-ejective), then $M$ is a CLS-module and some known results are generalized. Tercan [8] proved that if a module $M=M_{1}\oplus M_{2}$ where $M_{1}$ and $M_{2}$ are CS-modules such that $M_{1}$ is $M_{2}$-injective, then $M$ is a CS-module if and only if $Z_{2}(M)$ is a CS-module. Here we will show that Tercan's claim is not true.

A generalization of supplemented modules

Let $M$ be a left module over a ring $R$ and $I$ an ideal of $R$. $M$ is called an $I$-supplemented module (finitely $I$-supplemented module) if for every submodule (finitely generated submodule) $X$ of $M$, there is a submodule $Y$ of $M$ such that $X+Y=M$, $X\cap Y\subseteq IY$ and $X\cap Y$ is PSD in $Y$. This definition generalizes supplemented modules and $δ$-supplemented modules. We characterize $I$-semiregular, $I$-semiperfect and $I$-perfect rings which are defined by Yousif and Zhou [15] using $I$-supplemented modules. Some well known results are obtained as corollaries.

任意模的半素子模与实子模是素子模与实素子模的交集

Semiprime and real submodules of arbitrary modules are intersections of prime and real prime submodules

本文证明任意交换环上任意模的半素子模与实子模，分别是包含它们的素子模与实素子模的交集，推广了此前仅适用于有限生成模的相关结论。

关于经典1-吸收素子模

On classical 1-absorbing prime submodules

本文定义交换环上模的经典1-吸收素子模，研究其性质、张量积与融合重复中的相应子模，并刻画 m²=0 的局部环。

Modules in which pure submodule is essential in a direct summand

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.

On graded $A$-2-absorbing submodules of graded modules over graded commutative rings

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring, $M$ a graded $R$-module and $A\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $A$-2-absorbing submodules of $M$ as a generalization of graded 2-absorbing submodules and graded $A$-prime submodules of $M.$ We investigate some properties of this class of graded submodules.

A relationship between 2-primal modules and modules that satisfy the radical formula

The coincidence of the set of all nilpotent elements of a ring with its prime radical has a module analogue which occurs when the zero submodule satisfies the radical formula. A ring $R$ is 2-primal if the set of all nilpotent elements of $R$ coincides with its prime radical. This fact motivates our study in this paper, namely, to compare 2-primal submodules and submodules that satisfy the radical formula. A demonstration of the importance of 2-primal modules in bridging the gap between modules over commutative rings and modules over noncommutative rings is done and new examples of rings and m…

Krull维数为1的交换整环上的无挠模

Torsion-free Modules over Commutative Domains of Krull Dimension One

本研究探讨Krull维数1的交换整环上有限生成无挠模直和类的直和项封闭性，给出局部环、诺特环等情形的等价条件，分析h-局部环的局部化性质及逆命题成立的情形，并证明有限特征整环上可数生成模的同构类由亏格决定。

I-primary submodules

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.

Semimodules over commutative semirings and modules over unitary commutative rings

We study the so-called closed and splitting subsemimodules and submodules of a given semimodule or module, respectively. We describe lattices of subsemimodules and of closed subsemimodules and posets of splitting subsemimodules and submodules. In the case of modules a natural bijective correspondence between these posets and posets of projections is established.

一个定量 Hilbert 基定理与构造性 Krull 维数

A quantitative Hilbert's basis theorem and the constructive Krull dimension

本文利用构造性代数中的 $\alpha$-Noether 模概念，重新证明 Brookfield 关于 Noether 多项式环长度的结果，并给出域及整数上多元多项式环 Krull 维数上界的新构造性证明。

模的Möbius函数与薄表示

Möbius function for modules and thin representations

该研究聚焦模的子模偏序集的Möbius函数及有限性问题，给出半单模Möbius函数的自同态环计算方法，并详细探讨有界路代数表示的Möbius函数。

On modules of linear transformations

Let $D$ be a division ring, $\mathcal V$ and $ \mathcal W$ vector spaces over $D$, and ${\mathcal L(\mathcal V,\mathcal W)}$ the ${\mathcal L(\mathcal W)}$-${\mathcal L(\mathcal V)}$ bimodule of all linear transformations from $\mathcal V$ into $\mathcal W$. We prove some basic results about certain submodules of $\mathcal L(\mathcal V, \mathcal W)$. For instance, we show, among other results, that a right submodule (resp. left submodule) of ${\mathcal L(\mathcal V,\mathcal W)}$ is finitely generated whenever its image (resp. coimage) is finite-dimensional.

Modules with ascending chain condition on annihilators and Goldie modules

Using the concepts of prime module, semiprime module and the concept of ascending chain condition (ACC) on annihilators for an $R$-module $M$ . We prove that if \ $M$ is semiprime \ and projective in $σ\left[ M\right] $, such that $M$ satisfies ACC on annihilators, then $M$ has finitely many minimal prime submodules. Moreover if each submodule $N\subseteq M$ contains a uniform submodule, we prove that there is a bijective correspondence between a complete set of representatives of isomorphism classes of indecomposable non $M$-singular injective modules in $σ\left[ M\right] $ and the set of min…

Prime M-Ideals, M-Prime Submodules, M-Prime Radical and M-Baer's Lower Nilradical of Modules

Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, which coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M-m-system sets, M-n-system sets, M-prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M, called "prime M-ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this d…

分解为投射和稳定子模的直接和

Decompositions into a direct sum of projective and stable submodules

本文研究了在特定类中模块分解为投射子模和稳定子模的条件，并给出了分解失败的例子，包括无限生成或有限生成的模块。

一个度量完备且克鲁尔-施米茨空间的多参数持续性模块

A metrically complete and Krull--Schmidt space of multiparameter persistence modules

本文研究了多参数持续性模块的可观测范畴，证明其在度量和代数上具有完备性和唯一分解性，并展示了其在多参数持续性研究中的重要性。

Sum-Essential Graphs of Modules

The sum-essential graph $ \mathcal{S}_R(M) $ of a left $R$-module $M$ is a graph whose vertices are all nontrivial submodules of $M$ and two distinct submodules are adjacent iff their sum is an essential submodule of $M$. Properties of the graph $\mathcal{S}_R(M)$ and its subgraph $\mathcal{P}_R(M)$ induced by vertices which are not essential as submodules of $M$ are investigated. The interplay between module properties of $M$ and properties of those graphs is studied.

A Note on the Arrangement of Subgroups in the Automorphism Groups of Submodule Lattices of Free Modules

A complete description of subgroups in the general linear group over a semilocal ring containing the group of diagonal matrices was obtained by Z.I.Borewicz and N.A.Vavilov. It is shown in the present paper that a similar description holds for the intermediate subgroups of the group of all automorphisms of the lattice of right submodules of a free finite rank R-module over a simple Artinian ring containing the group consisting of those automorphisms which leave invariant an appropriate sublattice.

投射模的相似子模

Similar submodules of projective modules

研究投射模子模的相似关系，建立最大子模数量下界，并探讨其在矩阵环中的应用。

Krull-Gabriel dimension of domestic string algebras

We calculate, confirming a conjecture of Schröer, the Krull-Gabriel dimension of the category of modules over any domestic string algebra, as well as the Cantor-Bendixson rank of each point of its Ziegler spectrum. We also determine the topology on this space.

From Submodule Categories to the Stable Auslander Algebra

We construct two functors from the submodule category of a self-injective representation-finite algebra $Λ$ to the module category of the stable Auslander algebra of $Λ$. These functors factor through the module category of the Auslander algebra of $Λ$. Moreover they induce equivalences from the quotient categories of the submodule category modulo their respective kernels and said kernels have finitely many indecomposable objects up to isomorphism. Their construction uses a recollement of the module category of the Auslander algebra induced by an idempotent and this recollement determines a ch…

借助若干类型模刻画弱伪准素2-吸收子模

Some Characterizations of Weakly Pseudo Primary 2-Absorbing Submodules in Terms of some Types of Modules

本文研究弱伪准素2-吸收子模，利用忠实、非奇异、Z-正则及投射模刻画乘法模，并在附加条件下证明其子模剩余为相应理想。

Nil 模与子模的包络

Nil modules and the envelope of a submodule

本文研究交换幺环上子模的包络性质，证明半素根在包络升链中为不变量，满足根式的环中零子模包络生成幂等根并导出由 nil 模与既约模构成的挠理论，且 Noetherian 单列模满足半素根式。

A note on torsion modules with pure embeddings

We study Martsinkovsky-Russell torsion modules [MaRu20] with pure embeddings as an abstract elementary class. We give a model-theoretic characterization of the pure-injective and the $Σ$-pure-injective modules relative to the class of torsion modules assuming that the torsion submodule is a pure submodule. Our characterization of relative $Σ$-pure-injective modules strictly extends the classical charactetization of [GrJe76] and [Zim, 3.6]. We study the limit models of the class and determine when the class is superstable assuming that the torsion submodule is a pure submodule. As a corollary, …

Translation invariant linear spaces of polynomials

A set of polynomials $M$ is called a {\it submodule} of $\mathbb{C} [x_1, \dots, x_n ]$ if $M$ is a translation invariant linear subspace of $\mathbb{C} [x_1, \dots, x_n ]$. We present a description of the submodules of $\mathbb{C} [x,y]$ in terms of a special type of submodules. We say that the submodule $M$ of $\mathbb{C} [x,y]$ is an {\it L-module of order} $s$ if, whenever $F(x,y)=\sum_{n=0}^N f_n (x) \cdot y^n \in M$ is such that $f_0 =\ldots = f_{s-1}=0$, then $F=0$. We show that the proper submodules of $\mathbb{C} [x,y]$ are the sums $M_d +M$, where $M_d =\{ F\in \mathbb{C} [x,y] \colo…

Graded $r$-Submodules

Let $G$ be a group with identity $e$ and $R$ a commutative $G$-graded ring with a nonzero unity $1$. In this article, we introduce the concepts of graded $r$-submodules and graded special $r$-submodules, which are generalizations for the notion of graded r-ideals. For a nonzero $G$-graded $R$-module $M$, a proper graded $R$-submodule $K$ of $M$ is said to be graded $r$-submodule (resp., graded special $r$-submodule) if whenever $a\in h(R)$ and $x\in h(M)$ such that $ax\in K$ with $Ann_{M}(a)=\{0\}$ (resp., $Ann_{R}(x)=\{0\}$), then $x\in K$ (resp., $a\in (K:_{R}M)$). We study various propertie…

On Graded s-Prime Submodules

In this article, we introduce the concepts of graded $s$-prime submodules which is a generalization of graded prime submodules. We study the behavior of this notion with respect to graded homomorphisms, localization of graded modules, direct product, and idealization. We succeeded to prove the existence of graded $s$-prime submodules in the case of graded-Noetherian modules. Also, we provide some sufficient conditions for the existence of such objects in the general case, as well as, in the particular case of grading by $\mathbb{Z}$, a finite group, or a polycyclic-by-finite group, in addition…

欧几里得模、其直和的子模及强简化阶梯形

Euclidean modules, submodules of their direct sums and strong reduced echelon form

本文研究环上的欧几里得模及其直和子模，引入强简化阶梯形，并证明欧几里得模类广泛存在于多种经典代数构造中。

Generalizations of prime submodules over non-commutative rings

Throughout this paper, $R$ is an associative ring (not necessarily commutative) with identity and $M$ is a right $R$-module with unitary. In this paper, we introduce a new concept of $ϕ$-prime submodule over an associative ring with identity. Thus we define the concept as following: Assume that $S(M)$ is the set of all submodules of $M$ and $ϕ:S(M)\rightarrow S(M)\cup\{\emptyset\}$ is a function. For every $Y\in S(M)$ and ideal $I$ of $R,$ a proper submodule $X$ of $M$ is called $ϕ$-prime, if $YI\subseteq X$ and $YI\nsubseteqϕ(X),$ then $Y\subseteq X$ or $I\subseteq(X:_{R}M)$. Then we examine …

On Tight Submodules of Modules over Valuation Domains

This note offers an unusual approach of studying a class of modules inasmuch as it is investigating a subclass of the category of modules over a valuation domain. This class is far from being a full subcategory, it is not even a category. Our concern is the subclass consisting of modules of projective dimension not exceeding one, admitting only morphisms whose kernels and cokernels are also objects in this subclass. This class is still tractable, several features are in general simpler than in module categories, but lots of familiar properties are lost. A number of results on modules in this c…

On strongly harmonic and Gelfand modules

We introduce the notions of Strongly harmonic and Gelfand module, as a generalization of the well-known ring theoretic case. We prove some properties of these modules and we give a characterization via their lattice of submodules and their space of maximal submodules. It is also observed that, under some assumptions, the space of maximal submodules of a strongly harmonic module constitutes a compact Hausdorff space whose frame of open sets is isomorphic to the frame $Ψ(M)$ defined in [arXiv:1612.07407]. Finally, we mention some open questions that arose during this investigation.

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 …

广义迹子模与自同态环的中心

Generalized trace submodules and centers of endomorphism rings

本文推广经典迹映射以统一多种迹子模概念，并据此推广 Lindo 关于自同态环中心的计算，证明在维数1的温和条件下，该中心的典范模可由 M 关于 R 的典范模的迹子模给出。

希尔伯特零点定理与模和层的实零点定理

Hilbert's Nullstellensatz and the real Nullstellensatz for modules and sheaves

该研究证明了适用于有限生成模子模及凝聚层的希尔伯特零点定理与实零点定理的几何部分，补充了此前关于半素子模结构的结果，还建立了相关性质与局部化的相容性。