Jawad Y. Abuhlail
In this note we develop a coalgebraic approach to the study of solutions of linear difference equations over modules and rings. Some known results about linearly recursive sequences over base fields are generalized to linearly (bi)recursive (bi)sequences of modules over arbitrary commutative ground rings.
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It is shown that every commutative local ring of bounded module type is an almost maximal valuation ring.
Endomorphism rings of modules appear as the center of a ring, as the fix ring of ring with group action or as the subring of constants of a derivation. This note discusses the question whether certain *-prime modules (introduced by Bican et al.) have a prime endomorphism ring. Several conditions are presented that guarantee the primness of the endomorphism ring. The contours of a possible example of a *-prime module whose endomorphism ring is not prime are traced.
Let $A$ be a ring with minimum condition on principal right ideals. It is proved that $\aleph_0$-distributive right (left) $A$-modules coincide with Artinian (Noetherian) right (left) $A$-modules. Rings, over which all right modules are direct sums of $\aleph_0$-distributive coincide with rings of of finite representation type. Rings, whose right modules are semidistributive, coincide with Kawada rings, over basis rings of which all right modules are completely cyclic. The studies of Tuganbaev are supported by Russian Scientific Foundation, project 22-11-00052.
It is shown that each almost maximal valuation ring R such that every indecomposable injective module is countably generated, satisfies the following condition (C): each fp-injective module is locally injective. The converse holds if R is a domain. Moreover, it is proved that a valuation ring that sastisfies this condition (C) is almost maximal.
Seminoetherian Modules over Non-Primitive HNP rings
本文研究非原始HNP环上的半诺特模结构，揭示其完整性质。
It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings.
In this paper a simple right R-module S over a ring R is called hypersimple if its injective hull E(S) is cyclic, and a ring R is called right hypersimple if every simple right R-module is hypersimple. We initiate a study of these new notions, and revisit Osofsky's work on hypercyclic rings, i.e. rings whose cyclic right modules have cyclic injective hulls.
On Classical Prime Near-ring Modules
本文推广经典素性到近环模，定义并区分四种经典素模，证明零化子性质，并刻画R-理想的经典m-系统。
On Rings over Which All Modules Are Direct Sums of Distributive Modules
本文引入分配模、半分配模的定义，研究所有右模均为半分配模的环的相关性质，属于环与模理论领域的基础研究。
If R is a commutative ring, we prove that every finitely generated module has a pure-composition series with indecomposable factors and any two such series are isomorphic if and only if R is a Bezout ring and a CF-ring.
We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie and perfect torsion theories are differential. We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations t…
In this paper we compute the set of point modules of finitely semi-graded rings. In particular, from the parametrization of the point modules for the quantum affine n-space, the set of point modules for some important examples of non $\mathbb{N}$-graded quantum algebras is computed.
Categorical constructions on heaps and modules over trusses are considered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coproducts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. It is shown that the direct sum of two non-empty Abelian heaps is always infinite and isomorphic to the heap associated to the direct sums of the group retracts of both heaps and $\mathbb{Z}$. Direct sum is used to extend a given truss to a ring-type …
Classification of multiplication modules over multiplication rings with finitely many minimal primes
本文对具有有限多个极小素理想的乘法环上的乘法模进行分类，并用忠实、诺特、分配模刻画此类环，证明忠实诺特分配模与忠实乘法模在该条件下等价。
Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=\left(\begin{smallmatrix} A & 0 \\ U & B \\\end{smallmatrix}\right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over $T$, and discuss when a left $T$-module is strongly Gorenstein projective or strongly Gorenstein injective module.
It is shown that a local ring R of bounded module type is an almost maximal valuation ring if there exists a non-maximal prime ideal J such that R/J is a maximal valuation domain.
In this paper, the notion of strongly G_C-projective and injective modules is introduced, where C is a semidualizing module. Using these modules we can obtain a new characterization of G_C-projective and injective modules, similar to the one of projective modules by the free modules. We then define and study the global dimensions of rings relative to a semidualizing module C, and prove that the global G_C-projective dimension of a ring R is equal to the global G_C-injective dimension of R.
$\textbf{Theorem 1.3.}$ For a given ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\textbf{1)}$ Every non-singular right $A$-module $X$ which is is injective with respect to some essential right ideal of the ring $A$ is an injective module. $\textbf{2)}$ $A/G(A_A)$ is a right strongly semiprime ring. $\textbf{Theorem 1.4.}$ For a given ring $A$, the following conditions are equivalent. $\textbf{1)}$ $A$ is a right strongly semiprime ring. $\textbf{2)}$ Every right $A$-module which is injective with respect to some essential right ideal of the ring $A$, i…
We introduce and investigate ss-injectivity as a generalization of both soc-injectivity and small injectivity. A module M is said to be ss-N-injective (where N is a module) if every R-homomorphism from a semisimple small submodule of N into M extends to N. A module M is said to be ss-injective (resp. strongly ss-injective), if M is ss-R-injective (resp. ss-N-injective for every right R-module N). Some characterizations and properties of (strongly) ss-injective modules and rings are given. Some results of Amin, Yuosif and Zeyada on soc-injectivity are extended to ss-injectivity. Also, we provid…
An associative ring with 1 is said to be semilocal provided it is semisimple artinian modulo its Jacobson radical, that is, modulo its Jacobson radical it is isomorphic to a finite product of matrices over division rings. Modules with a semilocal endomorphism ring inherit some properties of semisimple modules. For example, they can be decomposed into a finite direct sum of indecomposable submodules, such decomposition is not unique but there are only a finite number of different ones, they cancel from direct sums and they satisfy the n-th root uniqueness property. A ring homomorphism is said t…
Recently, in a series of papers "simple" versions of direct-injective and direct-projective modules have been investigated. These modules are termed as "simple-direct-injective" and "simple-direct-projective", respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the l…
A weak basis of a module is a generating set of the module minimal with respect to inclusion. A module is said to be regularly weakly based provided that each of its generating sets contain a weak basis. In the paper we study (1) rings over which all modules are regularly weakly based, refining results of Nashier and Nichols, (2) regularly weakly based modules over Dedekind domains.
The rational hull of modules
本文利用相对稠密性质刻画环的极大右商环，研究模的有理包与自同态环的嵌入关系，并给出有理完备模的新刻画及相关等价和充分条件。
We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring which is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo-Gillespie-Hovey. It is also compatible with the notion of $n$-coherent rings introduced by Bravo-Perez: So a $0$-coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$-coherent AC-Gorenstein ring is precisely a Ding-Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of …
We introduce a complete radical formula for modules over non-commutative rings which is the equivalence of a radical formula in the setting of modules defined over commutative rings. This gives a general frame work through which known results about modules over commutative rings that satisfy the radical formula are retrieved. Examples and properties of modules that satisfy the complete radical formula are given. For instance, it is shown that a module that satisfies the complete radical formula is completely semiprime if and only if it is a subdirect product of completely prime modules. This g…
In this note we show that a ring R is left perfect if and only if every left R-module is weakly supplemented if and only if R is semilocal and the radical of the countably infinite free left R-module has a weak supplement.
$\textbf{Theorem 1.2.}$ 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. $\textbf{3)}$ $A=S\times T$, where $S$ is a right injective regular ring and $T$ is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients. $\textbf{Theorem 1.5.}$ For a ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\textbf{1)}$ $A/G(A_A)$ is a semiprime right Goldie ring. $\textbf{2)}$ …
It is proved, among other results, that a prime right nonsingular ring (in particular, a simple ring) $R$ is right self-injective if $R_R$ is invariant under automorphisms of its injective hull. This answers two questions raised by Singh and Srivastava, and Clark and Huynh. An example is given to show that this conclusion no longer holds when prime ring is replaced by semiprime ring in the above assumption. Also shown is that automorphism-invariant modules are precisely pseudo-injective modules, answering a recent question of Lee and Zhou. Furthermore, rings whose cyclic modules are automorphi…
Rings whose mininjective modules are injective
本文刻画了极小内射模均为内射的环，证明其等价于每个极小平坦模均投射的诺特环，并研究了Kasch、交换及拟Frobenius情形，以及上三角矩阵环的充要条件，还构造了贫乏模的新例子。
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 singly projective module is a content module if and only if any non-unit of $R$ is a zero-divisor and that each singly projective module is locally projective if and only if $R$ is self injective. Moreover, $R$ is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficie…
The Gelfand--Kirillov dimension has gained importance since its introduction as an tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand--Kirillov dimension of simple modules over certain simple rings of differential operators.
In this paper we establish a very close link (in terms of von Neumann's coordinatization) between regular modules introduced by Zelmanowitz, on one hand, and von Neumann regular rings, on the other hand: we prove that the lattice $\mathcal{L}^{fg}(M)$ of all finitely generated submodules of a finitely generated regular module $M$, over an arbitrary ring, can be coordinatized as the lattice of all principal right ideals of some von Neumann regular ring $S$.
Let $M$ be a module. A {\em $δ$-cover} of $M$ is an epimorphism from a module $F$ onto $M$ with a $δ$-small kernel. A $δ$-cover is said to be a {\em flat $δ$-cover} in case $F$ is a flat module. In the present paper, we investigate some properties of (flat) $δ$-covers and flat modules having a projective $δ$-cover. Moreover, we study rings over which every module has a flat $δ$-cover and call them {\em right generalized $δ$-perfect} rings. We also give some characterizations of $δ$-semiperfect and $δ$-perfect rings in terms of locally (finitely, quasi-, direct-) projective $δ$-covers and flat …
A generalization of uniserial modules and rings
本文引入弱单序列模与环的概念，证明其与简单阿廷环的等价条件，并刻画秩1无挠阿贝尔群及主理想整环上有限生成模的结构。
The aim of this paper is to introduce and study graded and filtered gamma rings and gamma modules. We prove that the filtered $Γ$-ring (module) is a generalization of the notion of graded ring (module). Also, we construct a graded $Γ$-ring from a filtered $Γ$-ring. We investigate some properties of graded and fileterd Gamma rings and Gamma modules. Finally we define and study the strongly graded gamma rings.
It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent. If, in addition, each finitely generated $R$-module has finite Goldie dimension, then localizations of finitely injective $R$-modules are finitely injective too. Moreover, if $R$ is a Prüfer domain of finite character, localizations of injective $R$-modules are injective.
Let $M_R$ be an injective right module over a ring $R$. The goal of this paper to prove that the endomorphism ring $S=(M_R)$ of $M$ is quasi-duo if and only if $M_R$ is square-free.
Let $R$ by a right coherent ring and $R$-Mod denote the category of left $R$-modules. We show that there is an abelian model structure on $R$-Mod whose cofibrant objects are precisely the Gorenstein flat modules. Employing a new method for constructing model structures, the key step is to show that a module is flat and cotorsion if and only if it is Gorenstein flat and Gorenstein cotorsion.
We prove that every non-finitely generated projective module over the integral group ring of a polycyclic-by-finite group G is free if and only if G is polycyclic.
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., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if R as a left (or as a right) R-module is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings. Finitely generated modu…
It is proved that the localization of an injective module E, over a valuation ring R, at a prime ideal J, is injective if J is not the subset of zero-divisors of R or if J or E is flat. It follows that localizations of injective modules over h-local Prüfer domains are injective too.
In this paper, we introduce and study the concept of CS-Rickart modules, that is a module analogue of the concept of ACS rings. A ring $R$ is called a right weakly semihereditary ring if every its finitly generated right ideal is of the form $P\oplus S,$ where $P_R$ is a projective module and $S_R$ is a singular module. We describe the ring $R$ over which $\mathrm{Mat}_n (R)$ is a right ACS ring for any $n \in \mathbb {N}$. We show that every finitely generated projective right $R$-module will to be a CS-Rickart module, is precisely when $R$ is a right weakly semihereditary ring. Also, we prov…
It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent. If, in addition, each finitely generated $R$-module has finite Goldie dimension, then localizations of finitely injective $R$-modules are finitely injective too. Moreover, if $R$ is a Prüfer domain of finite character, localizations of injective $R$-modules are injective.
Monogeny classes and epigeny classes have proved to be useful in the study of direct sums of uniserial modules and other classes of modules. In this paper, we show that they also turn out to be useful in the study of direct products.
We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantized Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained in [arXiv:0906.2930] for down-up algebras.
In this paper we discuss under which conditions cyclic essential extensions of simple modules over a differential operator ring R[z;d] are Artinian. In particular, we study the case when R is either d-simple or d-primitive. Furthermore, we obtain important results when R is an affine algebra of Kull dimension 2. As an application we characterize the differential operator rings C[x,y][z;d] for which cyclic essential extensions of simple modules are Artinian.
Generalized Bassian Modules over Non-primitive Dedekind Prime Rings
本文研究了非原始迪克因素数环上的广义巴西安模块性质，并探讨了其在环论中的应用。
Unifying some classical results on Artinian rings and modules
本文引入环R上左R-模类的测度概念，借此为Akizuki、Anderson、Hopkins和Levitzki关于Artinian环与模的经典定理及部分新结果提供了简短证明。
Let R be a ring, M a left R-module, I an infinite set, N either the direct sum or product of |I| copies of M, and E the endomorphism ring of N as a left R-module. In this note it is shown that E is not the union of a chain of |I| or fewer proper subrings, and also that given a generating set U for E as a ring, there exists a positive integer n such that every element of E is represented by a ring word of length at most n in elements of U.
We study injective hulls of simple modules over differential operator rings $R[θ; d]$, providing necessary conditions under which these modules are locally Artinian. As a consequence we characterize Ore extensions of $S=K[x][θ;σ, d]$ for $d$ a $K$-linear derivation and $σ$ a $K$-linear automorphism of $K[x]$ such that injective hulls of simple $S$-modules are locally Artinian.
In this paper we give a small review of some recent results of elementary equivalence of linear and algebraic groups and our last new results of elementary equivalence of categories of modules, endomorphism rings of modules, lattices of submodules of modules, and automorphism groups of modules.
There are well-known constructions relating ring epimorphisms and tilting modules. The new notion of silting module provides a wider framework for studying this interplay. To every partial silting module we associate a ring epimorphism which we describe explicitly as an idempotent quotient of the endomorphism ring of the Bongartz completion. For hereditary rings, this assignment is used to parametrise homological ring epimorphisms by silting modules. We further show that homological ring epimorphisms of a hereditary ring form a lattice which completes the poset of noncrossing partitions in the…
We study Gorenstein dimension and grade of a module $M$ over a filtered ring whose assosiated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G-dim$M\leq{G-dim gr}M$ and an equality ${\rm grade}M={\rm grade gr}M$, whenever Gorenstein dimension of ${\rm gr}M$ is finite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds a new viewpoint for studying filtered rings and modules. We ap…
We define pullback and separated presentations of modules over pullback rings, and, if the ring is a pullback of epimorphisms over a semisimple ring, an algorithm reducing such a presentation of a module to an $R$-diagram. The latter is the input for a classification algorithm of finitely generated modules over a pullback ring of two Dedekind domains. As an example we show how to obtain an $R$-diagram for homology of a chain complex of free modules over a $p$-pullback ring.
The behaviour under coarsening functors of simple, entire, or reduced graded rings, of free graded modules over principal graded rings, of superfluous monomorphisms and of homological dimensions of graded modules, as well as adjoints of degree restriction functors, are investigated.
In this paper we study (non-commutative) rings $R$ over which every finitely generated left module is a direct sum of cyclic modules (called left FGC-rings). The commutative case was a well-known problem studied and solved in 1970s by various authors. It is shown that a Noetherian local left FGC-ring is either an Artinian principal left ideal ring, or an Artinian principal right ideal ring, or a prime ring over which every two-sided ideal is principal as a left and a right ideal. In particular, it is shown that a Noetherian local duo-ring $R$ is a left FGC-ring if and only if $R$ is a right FG…
We consider the smallest triangulated subcategory of the unbounded derived module category of a ring that contains the injective modules and is closed under set indexed coproducts. If this subcategory is the entire derived category, then we say that injectives generate for the ring. In particular, we ask whether, if injectives generate for a collection of rings, do injectives generate for related ring constructions and vice versa. We provide sufficient conditions for this statement to hold for various constructions including recollements, ring extensions and module category equivalences.
Let $A$ be an Iwanaga-Gorenstein ring. Enomoto conjectured that a self-orthogonal $A$-module has finite projective dimension. We prove this conjecture for $A$ having the property that every indecomposable non-projective maximal Cohen-Macaulay module is periodic. This answers a question of Enomoto and shows the conjecture for monomial quiver algebras and hypersurface rings.
Firstly, we give a partial solution to the isomorphism problem for uniserial modules of finite length with the help of the morphisms between these modules over an arbitrary ring. Later, under suitable assumptions on the lattice of the submodules, we give a method to partially solve the isomorphism problem for uniserial modules over an arbitrary ring. Particular attention is given to the natural class of uniserial modules defined over algebras given by quivers.
A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring $R$ has the property that every $R$-module is a direct sum of cyclic modules if and only if $R$ is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in [M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi, Commutative Noetherian local rings whose ideals are direct sums of cyclic modules, J. Algebra 345 (2011) 257--265] the authors considered this question in the context of finite direct products of commutative Noetherian lo…
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