O. Broche, E. Jespers, C. Polcino Milies, M. Ruiz
Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let $\vp : RG\to RG$ denote the $R$-linear extension of an involution $\vp$ defined on $G$. An element $x$ in $RG$ is said to be $\vp$-antisymmetric if $\vp (x) = -x$. A characterization is given of when the $\vp$-antisymmetric elements of $RG$ commute. This is a completion of earlier work.
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Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let $ϕ_σ : RG\to RG$ denote the involution defined by $ϕ_σ (\sum r_{g}g) = \sum r_{g} σ(g) g^{-1}$, where $σ:G\to \{\pm 1\}$ is a group homomorphism (called an orientation morphism). An element $x$ in $RG$ is said to be antisymmetric if $ϕ_σ (x) =-x$. We give a full characterization of the groups $G$ and its orientations for which the antisymmetric elements of $RG$ commute.
For any ring \(R\), some characterizations are obtained for unit regular elements in a corner ring \(eRe\) in terms of unit regular elements in \(R\). \noindent {\bf Key Words}: von Neumann regular rings, unit regular rings, corner rings, idempotents \noindent {\bf AMS Classification}: 16A30
A commutative ring $R$ is J-stable provided that for any $a\not\in J(R)$, $R/aR$ has stable range one. A ring $R$ is called an elementary divisor ring if every $m\times n$ matrix over $R$ admits diagonal reduction. We prove that a J-stabe ring $R$ is an elementary divisor ring if and only if it is a Bezout ring.
We investigate and characterize several kinds of elements such as units, idempotents, von Neumann regular, $π$-regular and clean elements for skew PBW extensions over weak compatible rings. We also study the notions of Gelfand and Harmonic rings for these families of algebras. The results presented here extend those corresponding in the literature for commutative and noncommutative rings of polynomial type.
In this paper, we introduce a new class of rings whose elements are a sum of a central element and a nilpotent element, namely, a ring $R$ is called$CN$ if each element $a$ of $R$ has a decomposition $a = c + n$ where $c$ is central and $n$ is nilpotent. In this note, we characterize elements in $M_n(R)$ and $U_2(R)$ having CN-decompositions. For any field $F$, we give examples to show that $M_n(F)$ can not be a CN-ring. For a division ring $D$, we prove that if $M_n(D)$ is a CN-ring, then the cardinality of the center of $D$ is strictly greater than $n$. Especially, we investigate several kin…
We define morphic near-ring elements and study their behavior in regular near-rings. We show that the class of left morphic regular near-rings is properly contained between the classes of left strongly regular and unit regular near-rings.
In this paper we study and investigate concerning dependent elements of semiprime rings and prime rings R by using generalized derivation and derivation,when R admsit to satisfy some conditions,we give some results about that.
In this paper we consider the probability that two elements of a finite ring have product zero. We find bounds of this probability of a finite commutative ring with identity 1. The explicit computations for the ring $\mathbf{Z}_n$, the ring of integers modulo $n$, have been obtained.
We define the concepts of weakly precious and precious rings which generalize the notions of weakly clean and nil-clean rings. We obtain some fundamental properties of these rings. We also obtain certain subclasses of such rings and then offer new kinds of weakly clean rings and nil-clean rings.
This paper, we consider some properties of rings via q-potent and periodic elements. In this paper we give some results of rings in which every element is a sum of an idempotent and a q-potent that commute; periodic rings and k-potent elements of algebras.
Rank of elements of general rings in connection with unit-regularity
该研究定义了一般含单位元环中元素的秩，给出半素环中秩的可逆元刻画，并证明含单位元半素环基座的元素均为单位正则元。
Complemented elements in commutative rings
本文将补环的概念推广到未必约化的环，并探讨其与已知环类的关系。
We have results about the centralizer.
An element of a ring R is called clean if it is the sum of an idempotent and a unit. A ring R is called clean if each of its element is clean. An element r \in R called regular if r = ryr for some y \in R. The ring R is regular if each of its element is regular. In this paper we define a ring is r-clean if each of its elements is the sum of a regular and an idempotent element. We give some relations between r-clean and clean rings. Finally we investigate some properties of r-clean rings.
Let $R$ be a commutative ring of characteristic zero and $G$ an arbitrary group. In the present paper we classify the groups $G$ for which the set of symmetric elements with respect to the classical involution of the group ring $RG$ is Lie metabelian.
Rings Whose Non-Invertible Elements Are Uniquely Strongly Clean
本文定义并详细研究了GUSC环，即非可逆元素唯一强清洁的环，推广了USC环和GUC环的概念。
In this article several properties of the inverse along an element will be studied in the context of unitary rings. New characterizations of the existence of this inverse will be proved. Moreover, the set of all invertible elements along a fixed element will be fully described. Futhermore, commuting inverse along an element will be characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) will be also considered.
We prove that an integral Jacobson radical ring is always nil, which extends a well known result from algebras over fields to rings. As a consequence we show that if every element x of a ring R is a zero of some polynomial p_x with integer coefficients, such that p_x(1)=1, then R is a nil ring. With these results we are able to give new characterizations of the upper nilradical of a ring and a new class of rings that satisfy the Köthe conjecture, namely the integral rings.
In this paper, we mainly give characterizations of EP elements in terms of equations. In addition, a related notion named a central EP element is defined and investigated. Finally, we focus on characterizations of a generalized EP element, i.e. *-DMP element.
In this paper, we investigate *-DMP elements in $*$-semigroups and $*$-rings. The notion of *-DMP element was introduced by Patrício in 2004. An element $a$ is *-DMP if there exists a positive integer $m$ such that $a^{m}$ is EP. We first characterize *-DMP elements in terms of the \{1,3\}-inverse, Drazin inverse and pseudo core inverse, respectively. Then, we give the pseudo core decomposition utilizing the pseudo core inverse, which extends the core-EP decomposition introduced by Wang for matrices to an arbitrary $*$-ring; and this decomposition turns to be a useful tool to characterize *-DM…
On Reversible Rings, Their Generalization and Quasi-Idempotent Elements
本文证明可逆环及其相关环类关于幂等元的刻画可推广至拟幂等元，保持性质不变。
A commutative ring R is locally stable provided that for any $a,b\in R$ such that $aR+bR=R$, there exist some $y\in R$ such that $R/(a+by)R$ has stable range 1.For a Bezout ring $R$, we prove that $R$ is an elementary divisor ring if and only if $R$ is locally stable if and only if $R$ has neat range 1.
In \cite{C, LCC}, it is proven that if an element $ab$ in a ring is (generalized) Drazin invertible, then so is $ba$. In this paper, we give a new and short proof of it in an effective manner. In particular, we show that if $ab$ is strongly clean, then so is $ba$. Consequently, we see that if $1-ab$ is strongly clean, then so is $1-ba$. Also, some characterizations are obtained for some certain elements in a corner ring. It is shown that for an idempotent $e$ and any arbitrary element $a$ in a ring, $ea+1-e$ is Drazin invertible if and only if $eae$ is Drazin invertible.
The Sum of EP elements in a ring with involution
该研究针对带对合环中EP元素和的core逆存在性问题，给出充要条件，并将其应用于含EP子块的分块复矩阵core逆条件的推导。
Regarding the question of how idempotent elements affect reversible property of rings, we study a version of reversibility depending on idempotents. In this perspective, we introduce {\it right} (resp., {\it left}) {\it $e$-reversible rings}. We show that this concept is not left-right symmetric. Basic properties of right $e$-reversibility in a ring are provided. Among others it is proved that if $R$ is a semiprime ring, then $R$ is right $e$-reversible if and only if it is right $e$-reduced if and only if it is $e$-symmetric if and only if it is right $e$-semicommutative. Also, for a right $e…
Rings whose Nil-Clean and Clean Elements are Uniquely Nil-Clean
研究每个 nil-clean 或 clean 元素均为 uniquely nil-clean 的环，证明了 abelian rings 与此类环的类重合，并对 semipotent 情形给出了完全刻画。
{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.
Element absorb Topology on rings
本文在非交换环上定义由元素右乘作用诱导的元素吸收拓扑，并利用幂等元情形给出 Pierce 分解的拓扑描述。
We present new characterizations of the rings in which every element is the sum of two idempotents and a nilpotent that commute, and the rings in which every element is the sum of two tripotents and a nilpotent that commute. We prove that such rings are completely determined by the additive decompositions of their square elements.
Unipotent orbits of elements in a quaternion ring of odd order
本文研究了四元数环中奇数阶元素的单幂轨道，并通过矩阵环的还原方法对单幂相似类进行分类。
One sided a_idempotent, one sided a_equivalent and SEP elements in a ring with involution
本文通过引入单边a_幂等元与单边a_等价元概念，在群可逆且MP_可逆条件下刻画了带对合环中SEP元素的等价特征。
Let $R$ be a unital ring with involution. We first show that the EP elements in $R$ can be characterized by three equations. Namely, let $a\in R$, then $a$ is EP if and only if there exists $x\in R$ such that $(xa)^{\ast}=xa$, $xa^{2}=a$ and $ax^{2}=x.$ It is well known that all EP elements in $R$ are core invertible and Moore-Penrose invertible. We give more equivalent conditions for a core (Moore-Penrose) invertible element to be an EP element. Finally, the EP elements are characterized in terms of $n$-EP property, which is a generalization of bi-EP property.
Notes on Reversed Products of Two Elements in Rings
本文受Cline公式启发，用群可逆和EP元素刻画可逆环与$\ast$-可逆环，证明两个等价条件，并构造反例区分环类及讨论包含关系。
In this paper, the notion of a uniformly distributed systems of elements on the variety of metabelian Lie algebras is introduced. This notion is analogous to one of a measure preserving systems of elements on group varieties. As the main result of the paper it was shown that on the variety of metabelian Lie algebras a system of elements is primitive iff it is uniformly distributed.
We construct an example of a unit-regular ring which is not strongly clean, answering an open question of Nicholson. We also characterize clean matrices with a zero column, and this allows us to describe an interesting connection between unit-regular elements and clean elements. It is also proven that given an element $a$ in a ring $R$, if $a,a^2,\ldots, a^k$ are all regular elements in $R$ (for some $k\geq 1$), then there exists $w\in R$ such that $a^{i}w^{i}a^{i}=a^{i}$ for $1\leq i\leq k$, and a similar statement holds for unit-regular elements. The paper ends with a large number of example…
Let $R$ be an associative ring. We define a subset $S_{R}^{a}$, where $a\in R$ of $R$ as $S_{R}^{a}=\{b\in R \mid aRb=(0)\}$. Then, the set $P_{R} = \bigcap_{a\in R} S_{R}^{a}$ call it the source of primeness of $R$. We first examine some basic properties of the subset $P_{R}$ in any ring $R$, and properties of idempotent elements, nilpotent elements, zero divisor elements and identity elements. And we investigated the properties of the elements of the set source of primeness with the help of these elements.
We use George Bergman's recent normal form for universally adjoining an inner inverse to show that, for general rings, a nilpotent regular element $x$ need not be unit-regular. This contrasts sharply with the situation for nilpotent regular elements in exchange rings (a large class of rings), and for general rings when all powers of the nilpotent element $x$ are regular.
In this paper, we prove that the semigroups of invertible matrices with nonnegative elements over linearly oredered associative rings are elementarily equivalent if and only if the matrices have the same dimension and the rings are elementarily equivalent as ordered rings.
Strongly clean ring elements that are one-sided inverses
针对“每个强clean环是否都是Dedekind有限环”这一长期开放问题，本文构造了一个环例子，其中两个强clean元素互为单侧逆但非双侧逆，为该问题提供否定答案的可能，并讨论强化结果的方法及相关观察。
Suppose that R is an ordered ring, G_n(R) is a subsemigroup of $GL_n(R)$, consisting of all matrices with nonnegative elements. A.V. Mikhalev and M.A. Shatalova described all automorphisms of G_n(R), where R is a linearly ordered skewfield and n>1. E.I. Bunina and A.V. Mikhalev found all automorphisms of G_n(R), if R is an arbitrary linearly ordered associative ring with 1/2, n>2. In this paper we describe automorphisms of G_n(R), if R is a commutative partially ordered ring, containing Q, n>2.
A ring is SSP if the sum of two direct summands is a direct summand. A ring has internal cancellation if every its (von Neumann) regular elements are unit-regular. We show that in an SSP ring having internal cancellation, any regular element is special clean. Our main results also imply that for any SSP ring internal cancellation and idempotent stable range $1$ coincide with each other. Internal cancellation over SSP was then characterized by special clean elements.
It was proved independently by both Wolfson [An ideal theoretic characterization of the ring of all linear transformations, Amer. J. Math. 75 (1953), 358-386] and Zelinsky [Every Linear Transformation is Sum of Nonsingular Ones, Proc. Amer. Math. Soc. 5 (1954), 627-630] that every linear transformation of a vector space $V$ over a division ring $D$ is the sum of two invertible linear transformations except when $V$ is one-dimensional over $\mathbb Z_2$. This was extended by Khurana and Srivastava [Right self-injective rings in which each element is sum of two units, J. Algebra and its Appl., V…
New partial orders on Rickart *-rings
研究在Rickart *-环上通过附加条件定义新偏序，证明元素下集与自伴幂等元子集序同构，刻画相关元素，分析正则环中元素对上下确界情况并扩展到正则Baer *-环子集。
We study the Drazin inverses of the sum and product of two elements in a ring. For Drazin invertible elements $a$ and $b$ such that $a^2b=aba$ and $b^2a=bab$, it is shown that $ab$ is Drazin invertible and that $a+b$ is Drazin invertible if and only if $1+a^Db$ is Drazin invertible. Moreover, the formulae of $(ab)^D$ and $(a+b)^D$ are presented. Thus, a generalization of the main result of Zhuang, Chen et al. (Linear Multilinear Algebra 60 (2012) 903-910) is given.
Armendariz ring property via idempotent elements
本文推广Armendariz环，引入右e-Armendariz环概念，证明其无左右对称性并给出例子，核心结论是环为右e-Armendariz环当且仅当e为左半中心元且角环eRe是Armendariz环。
On Quasi-Nil Clean Rings
本文引入拟幂零清洁环的概念，其中每个元素可表示为拟幂等元与幂零元之和，并研究其结构性质、与现有环类的关系及交换环的拟布尔商刻画。
The probability that the product of k elements in a finite ring is zero
该文研究有限交换环中随机k个元素乘积为零的概率$zp_{_k}(R)$，推导其紧界、确定最大值并分类特定概率范围内的环。
Rings in which one-sided strongly $π$-regular elements are strongly $π$-regular
本文研究了Dedekind-有限环中单边强π-正则元素的性质，并证明了该条件在左右对称性上的不对称性。
In this paper we study the probability that the commutator of a randomly chosen pair of elements, one from a subring of a finite ring and other from the ring itself equals to a given element of the ring.
We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and Jordan and generalizations thereof. We recall arithmetical invariants for the study of non-unique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and clas…
In the current article we study structure of a Chevalley group $G(R)$ over a commutative ring $R$. We generalize and improve the following results: (1) standard, relative, and multi-relative commutator formulas; (2) nilpotent structure of [relative] $K_1$; (3) bounded word length of commutators. To this end we enlarge the elementary group, construct a generic element for the extended elementary group, and use localization in the universal ring. The key step is a construction of a generic element for a principle congruence subgroup, corresponding to a principle ideal.
An element of a ring $R$ is strongly $P$-clean provided that it can be written as the sum of an idempotent and a strongly nilpotent element that commute. A ring $R$ is strongly $P$-clean in case each of its elements is strongly $P$-clean. We investigate, in this article, the necessary and sufficient conditions under which a ring $R$ is strongly $P$-clean. Many characterizations of such rings are obtained. The criteria on strong $P$-cleanness of $2\times 2$ matrices over commutative local rings are also determined.
Let R be a 2 torsion free semiprime ring and d a nonzero derivation. Further let A = O(R) be the orthogonal completion of R and B = B(C) the Boolean ring of C where C be the extended centroid of R. We show that if a[[d(x),x]^n- [y, d(y)]^m]^t = 0 such that a in R for all x, y in R, where m, n, t > 0 are fixed integers, then there exists an idempotent e in B such that eA is a commutative ring and d induce a zero derivation on (1-e)A.
Generalized $t$-Fine and Quasi $t$-Fine Rings
本文引入广义$t$-精细环与广义拟$t$-精细环，用挠单位元与（拟）幂零元分解Jacobson根外元素，刻画其性质并研究矩阵环、群环等上的表现。
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。
Motivated by the concept of weakly clean rings, we introduce the concept of weakly $r$-clean rings. We define an element $x$ of a ring $R$ as weakly $r$-clean if it can be expressed as $x=r+e$ or $x=r-e$ where $e$ is an idempotent and $r$ is a regular element of $R$. If all the elements of $R$ are weakly $r$-clean then $R$ is called a weakly $r$-clean ring. We discuss some of its properties in this article. Also we generalise this concept of weakly $r$-clean ring to weakly $g(x)$-$r$-clean ring, where $g(x) \in C(R)[x]$ and $C(R)$ is the centre of the ring $R$. Finally we introduce the concept…
Invertibility of Anticommutator and Commutators of Higher Degree of $n$-potent Elements
研究n - 幂零元的高阶交换子和反交换子的可逆性，通过在环的自然条件下进行分析，关联其与高阶同类的可逆性，并探讨在环扩张中该可逆性从基环继承的情况。
Let $R$ be a unital ring with involution.In this paper, several new necessary and sufficient conditions for the existence of the Moore-Penrose inverse of an element in a ring $R$ are given.In addition, the formulae of the Moore-Penrose inverse of an element in a ring are presented.
The core inverse for a complex matrix was introduced by Baksalary and Trenkler. Rakić, Dinčić and Djordjević generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible, in this paper, we will answer this question. We will use three equations to characterize the core inverse of an element. That is, let $a, b\in R$, then $a\in R^{\tiny\textcirc…
An element $a$ of a ring $R$ is called \emph{strongly $J$-clean} provided that there exists an idempotent $e\in R$ such that $a-e\in J(R)$ and $ae=ea$. A ring $R$ is \emph{strongly $J$-clean} in case every element in $R$ is strongly $J$-clean. In this paper, we investigate strong $J$-cleanness of $M_2(R;s)$ for a local ring $R$ and $s\in R$. We determine the conditions under which elements of $M_2(R;s)$ are strongly $J$-clean.
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