When $δ$-semiperfect rings are semiperfect

拓扑半完全拓扑环

Topologically semiperfect topological rings

本文定义并刻画拓扑半完全拓扑环，揭示其与局部自同态环模直和的关系，分析拓扑 Jacobson 根、商环、投射覆盖及幂等元提升等结构性质。

单模的自同态环与块分解

Endomorphism rings of simple modules and block decomposition

本文利用左右诺特半完全环不可分解性的特征刻画，通过Ext群研究单模，证明不可分解半完全环的单模基数及自同态环特征的一致性，并在拟Frobenius环情形下强化了结果。

Rings over which every module has a flat $δ$-cover

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 …

On Semilocal Modules and Rings

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…

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.

ADS modules

We study the class of ADS rings and modules introduced by Fuchs. We give some connections between this notion and classical notions such as injectivity and quasi-continuity. A simple ring R such that R is ADS as a right R-module must be either right self-injective or indecomposable as a right R-module. Under certain conditions we can construct a unique ADS hull up to isomorphism. We introduce the concept of completely ADS modules and characterize completely ADS semiperfect right modules as direct sum of semisimple and local modules.

由silting模生成的TTF类

TTF classes generated by silting modules

研究环上模范畴中TTF类成为silting类的条件，通过幂等理想与TTF类的对应关系，得出R/I为silting模的充要条件，证明I是投射模迹时R/I为silting模且逆命题对半完全环等成立。

相对大投射模及其在直和分解中的应用

Relatively big projective modules and their applications to direct sum decompositions

本文扩展了可数生成投射模相对于迹理想总是相对大的环的例子列表，证明其包括满足多项式恒等式的右noetherian环及$h$-局部整环上的局部半完备无挠代数，并以此完整刻画了Krull维数为1的$h$-局部整环上有限生成无挠模直和项的分解性质。

Rings close to periodic with applications to matrix, endomorphism and group rings

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not $R$ being nil-clean implies that $\mathbb{M}_n(R)$ is nil-clean for all $n\geq 1$ is paralleling to the corresponding implication for (Abelian, local) periodic rings. Besides, we study when the endomorphism ring $\mathrm{E}(G)$ of an Abelian group $G$ is periodic. Concretely, we establish that $\mathrm{E}(G)$ is periodic exactly when $G$ is finite as well as we find a complete necessary and sufficient condition when t…

Rings of invariants of finite groups when the bad primes exist

Let R be a ring (not necessarily with 1) and G be a finite group of automorphisms of R. The set B(R, G) of primes p such that p | |G| and R is not p-torsion free, is called the set of bad primes. When the ring is |G|-torsion free, i.e., B(R, G) is empty set, the properties of the rings R and R^G are closely connected. The aim of the paper is to show that this is also true when B(R, G) is not empty set under natural conditions on bad primes. In particular, it is shown that the Jacobson radical (resp., the prime radical) of the ring R^G is equal to the intersection of the Jacobson radical (resp.…

由n项大倾斜复形导出的导出范畴的重排

Recollements of derived categories from $n$-term big tilting complexes

针对环A上的n项大倾斜复形T，构造B-Mod的扩张闭正合子范畴ℰ，证明导出范畴DB可由Dℰ和DA重排，还得到ℰ的d-对称性、阿贝尔性条件，将两项经典情形推广到任意有限项，构造了n≥2时的非紧大倾斜复形。

无限环何时具有有限压缩交换图？

When does an infinite ring have a finite compressed commuting graph?

本研究证明任意无限环都有无限非单位压缩交换图，以环的半直积为工具分类了具有有限单位压缩交换图的无限单位环，并由此分类了仅含有限个单位子环的无限单位环。

When is every non central-unit a sum of two nilpotents?

A ring is said to satisfy the $2$-nil-sum property if every non central-unit is the sum of two nilpotents. We prove that a ring satisfies the $2$-nil-sum property iff it is either a simple ring with the $2$-nil-sum property or a commutative local ring with nil Jacobson radical, and we provide an example of a simple ring with the $2$-nil-sum property that is not commutative. Moreover, a simple right Goldie ring has the $2$-nil-sum property iff it is a field.

Rings in which every element is either a sum or a difference of a nilpotent and an idempotent

{Generalizing the notion of nil cleanness from \cite{D13}, in parallel to \cite{DM14}, we define the concept of {\it weak nil cleanness} for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition theorem of a weakly nil-clean ring is obtained. It is completely characterized when an abelian ring is {\it weakly nil-clean}.} It is also completely determined when a matrix ring over a division ring is weakly nil-clean.

When is an origami set a ring?

Starting with a flat sheet of paper, points can be constructed as the intersection of two folds. The set of constructible points clearly depends on which folds are admissible. In this paper, we study the situation where a fold is admissible if its slope is admissible and it contains an already constructed or a generator point. We give an explicit characterization of the set of constructible points. We also state several criteria for this set to be a ring. This answers questions originally raised by Erik Demaine and discussed by Butler et al. and Buhler et al.

On cyclic essential extensions of simple modules over differential operator rings

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.

Filtered skew derivations on simple artinian rings

Given a complete, positively filtered ring $(R,f)$ and a compatible skew derivation $(σ,δ)$, we may construct its skew power series ring $R[[x;σ,δ]]$. Due to topological obstructions, even if $δ$ is an \emph{inner} $σ$-derivation, in general we cannot ``untwist" it, i.e. reparametrise to find a filtered isomorphism $R[[x; σ, δ]] \cong R[[x'; σ]]$, as might be expected from the theory of skew polynomial rings; similarly when $σ$ is an inner automorphism. We find general conditions under which it is possible to untwist the multiplication data, and use this to analyse the structure of $R[[x;σ,δ]]…

On a special presentation of matrix algebras

Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $n\times n$ matrix ring, so $R\cong M_{n}(S)$ for some ring $S$, if and only if it contains a set of $n\times n$ matrix units $\{e_{ij}\}_{i,j=1}^n$. A more recent and less known result states that a ring $R$ is a complete $(m+n)\times(m+n)$ matrix ring if and only if, $R$ contains three elements, $a$, $b$, and $f$, satisfying the two relations $af^m+f^nb=1$ and $f^{m+n}=0$. In many instances the two elements $a$ and $b$ can be replaced by appropria…

Relative Gorenstein dimensions over triangular matrix rings

Let $A$ and $B$ be rings, $U$ a $(B,A)$-bimodule and $T=\begin{pmatrix} A&0\\U&B \end{pmatrix}$ the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over $T$ using the corresponding ones over $A$ and $B$. We show that when $U$ is relative (weakly) compatible we are able to describe the structure of $G_C$-projective modules over $T$. As an application, we study when a morphism in $T$-Mod has a special $G_CP(T)$-precover and when the class $G_CP(…

Endomorphism rings via minimal morphisms

We prove that if $u:K \rightarrow M$ is a left minimal extension, then there exists an isomorphism between two subrings, $\textrm{End}_R^M(K)$ and $\textrm{End}_R^K(M)$ of $\textrm{End}_R(K)$ and $\textrm{End}_R(M)$ respectively, modulo their Jacobson radicals. This isomorphism is used to deduce properties of the endomorphism ring of $K$ from those of the endomorphism ring of $M$ in certain situations such us when $K$ is invariant under endomorphisms of $M,$ or when $K$ is invariant under automorphisms of $M$.

Coherent rings of differential operators

We consider the following question: When are rings of differential operators coherent? If $A$ is a finitely generated smooth domain over a field $k$ of characteristic $0$, then the ring $D$ of differential operators on $A$ is a Noetherian ring and a finitely generated $k$-algebra. However, when $k$ has characteristic $p > 0$ or when $A$ is singular, this is no longer true. In fact, Bernstein, Gelfand and Gelfand showed that for the cubic cone $A = k[x,y,z]/(x^3 + y^3 + z^3)$, the ring $D$ is neither Noetherian nor finitely generated if $k$ has characteristic $0$, and the same is true for the p…

When mutually subisomorphic Baer modules are isomorphic

The Schröder-Bernstein Theorem for sets is well known. The question of whether two subisomorphic algebraic structures are isomorphic to each other, is of interest. An $R$-module $M$ is said to satisfy the Schröder-Bernstein (or SB) property if any pair of direct summands of $M$ are isomorphic provided that each one is isomorphic to a direct summand of the other. A ring $R$ (with an involution $\star$) is called a Baer (Baer $\star$-)ring if the right annihilator of every nonempty subset of $R$ is generated by an idempotent (a projection). It is clear that every Baer $\star$-ring is a Baer ring…

Green ring of the category of weight modules over the Hopf-Ore extensions of group algebras

In this paper, we continue our study of the tensor product structure of category $\mathcal W$ of weight modules over the Hopf-Ore extensions $kG(χ^{-1}, a, 0)$ of group algebras $kG$, where $k$ is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of $χ$ and $χ(a)$ are different. Then we describe the Green ring $r(\mathcal W)$ of the tensor category $\mathcal W$. It is shown that $r(\mathcal W)$ is isomorphic to the polynomial algebra over the group ring $\math…

On the weak krull symmetry of a noetherian ring

We define when a noetherian ring R is called a right ( or a left) weakly krull symmetric ring . We then prove that if R is a right ( or a left ) krull homogenous ring then R is a right ( or a left ) weakly krull symmetric ring . This result modifies the main result of [2] . The key terms introduced in this paper are of independent interest .

非单位是单位倍数的Δ(R)元素的环

Rings whose Non-Units are a Unit Multiple of an Element from $\sqrt{Δ(R)}$

本文研究了$U\sqrtΔ$-环的性质，证明其不可分解性和德克因有限性，并探讨了多项式环、幂级数环及群环等在特定条件下的$U\sqrtΔ$-环特性。

Maximal Graded Orders over Crystalline Graded Rings

Crystalline graded rings are generalizations of certain classes of rings like generalized twisted group rings, generalized Weyl algebras, and generalized skew crossed products. When the base ring is a commutative Dedekind domain, two constructions are given for producing maximal graded orders. On the way, a new concept appears, so-called, spectrally twisted group. Some general properties of it are studied. At the end of the paper several examples are considered.

When the group ring of a finite group over a field is serial

We make a list of finite simple groups whose group rings over a given field are serial.

Simple Cuntz-Pimsner rings

Necessary and sufficient conditions for when every non-zero ideal in a relative Cuntz-Pimsner ring contains a non-zero graded ideal, when a relative Cuntz-Pimsner ring is simple, and when every ideal in a relative Cuntz-Pimsner ring is graded, are given. A "Cuntz-Krieger uniqueness theorem" for relative Cuntz-Pimsner rings is also given and condition (L) and condition (K) for relative Cuntz-Pimsner rings are introduced.

Symmetric units in integral group rings

In this paper, we study the question of when the symmetric units in an integral group ring ZG form a multiplicative group. When G is periodic, necessary and sufficient conditions are given for this to occur.

Clean group rings over localizations of rings of integers

A ring $R$ is said to be clean if each element of $R$ can be written as the sum of a unit and an idempotent. In a recent article (J. Algebra, 405 (2014), 168-178), Immormino and McGoven characterized when the group ring $\mathbb Z_{(p)}[C_n]$ is clean, where $\mathbb Z_{(p)}$ is the localization of the integers at the prime $p$. In this paper, we consider a more general setting. Let $K$ be an algebraic number field, $\mathcal O_K$ be its ring of integers, and $R$ be a localization of $\mathcal O_K$ at some prime ideal. We investigate when $R[G]$ is clean, where $G$ is a finite abelian group, a…

Symmetry on rings of differential operators

If $k$ is a field and $R$ is a commutative $k$-algebra, we explore the question of when the ring $D_{R|k}$ of $k$-linear differential operators on $R$ is isomorphic to its opposite ring. Under mild hypotheses, we prove this is the case whenever $R$ Gorenstein local or when $R$ is a ring of invariants. As a key step in the proof we show that in many cases of interest canonical modules admit right $D$-module structures. After this work was completed we realized that some of our results were already proved in higher generality by Yekutieli, albeit using more sophisticated methods.

Homotopy formulas for cyclic groups acting on rings

The positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm-one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff.

Zero-Divisor Graphs of Quotient Rings

The compressed zero-divisor graph $Γ_C(R)$ associated with a commutative ring $R$ has vertex set equal to the set of equivalence classes $\{ [r] \mid r \in Z(R), r \neq 0 \}$ where $r \sim s$ whenever $ann(r) = ann(s)$. Distinct classes $[r],[s]$ are adjacent in $Γ_C(R)$ if and only if $xy = 0$ for all $x \in [r], y \in [s]$. In this paper, we explore the compressed zero-divisor graph associated with quotient rings of unique factorization domains. Specifically, we prove several theorems which exhibit a method of constructing $Γ(R)$ for when one quotients out by a principal ideal, and prove suf…

当每个S-平坦模都是（平坦）投射模时

When every S-flat module is (flat) projective

本文研究含单位元交换环中S-平坦模均为平坦或投射模的环类，给出两类环的刻画，定义S-完全环，举例区分三类环并研究S-完全性的环构造传递性。

可逆元为弱幂清洁的环

Rings Whose Invertible Elements Are Weakly Nil-Clean

本文研究所有可逆元均为弱幂清洁的环（UWNC环），证明三角矩阵环UWNC当且仅当底环UNC，并刻画了UWNC 2-素环及多项式环的UWNC性质。

具有类清洁性质的环：自同态、矩阵与结构定理

Rings with Clean-Like Properties: Endomorphism, Matrix and Structural Theorems

研究任意环、阿贝尔群自同态环和有限域上矩阵环的三种类清洁性质，给出弱强k-零清洁、拟2-零清洁和拟3-零清洁的判定条件，改进了多个已有结果。

Chain conditions for rings with enough idempotents with applications to category graded rings

We obtain criteria for when a ring with enough idempotents is left/right artinian or noetherian in terms of local criteria defined by the associated complete set of idempotents for the ring. We apply these criteria to object unital category graded rings in general and, in particular, to the class of skew category algebras. Thereby, we generalize results by Nastasescu-van Oystaeyen, Bell, Park and Zelmanov from the group graded case to groupoid, and in some cases category, gradings.

Generalized prime ideals of the rings C(X,Y) and the quasi-components of X

In the set of continuous functions C(X,Y) where Y has a topology close to being discrete, there is an equivalence relation on X which characterizes the quasi-components of X. If Y satisfies weak algebraic conditions with a single binary operation then a stable set of functions forms an object generalizing an ideal of a ring. Calling such sets ideals there is a concept of a prime ideal. The ideal of functions vanishing on a quasi-component are prime ideals of C(X,Y). If Y has a zero set that is open then these prime ideals are min- max implying that when Y is a ring all of the prime ideals of C…

环ℤ/nℤ的Marshall商

Marshall Quotients of the Rings $\mathbb Z/n\mathbb Z$

该文研究模n整数环的Marshall商，给出其显式描述并分类结构性质，分析含零的可逆类子集，为相关代数问题提供有限测试例族。