Anvar Imomkulov
In this paper we construct some families of three-dimensional evolution algebras which satisfies Chapman-Kolmogorov equation. For all of these chains we study the behavior of the baric property, the behavior of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.
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We describe absolute nilpotent and some idempotent elements of an $n$- dimensional evolution algebra corresponding to two permutations and we decompose such algebras to the direct sum of evolution algebras corresponding to cycles of the permutations.
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.
A ring $R$ is a UU ring if every unit is unipotent, or equivalently if every unit is a sum of a nilpotent and an idempotent that commute. These rings have been investigated in Călugăreanu \cite{C} and in Danchev and Lam \cite{DL}. In this paper, two generalizations of UU rings are discussed. We study rings for which every unit is a sum of a nilpotent and an idempotent, and rings for which every unit is a sum of a nilpotent and two idempotents that commute with one another.
Decomposition of a square matrix into a product of a nilpotent matrix and an involutive or idempotent matrix
本研究以不变因子为依据，完整刻画了交换域上n阶方阵可分解为幂零矩阵与对合矩阵或幂等矩阵乘积的条件。
A ring R is a strongly 2-nil-clean if every element in R is the sum of two idempotents and a nilpotent that commute. A ring R is feebly clean if every element in R is the sum of two orthogonal idempotents and a unit. In this paper, strongly 2-nil-clean rings are studied with an emphasis on their relations with feebly clean rings. This work shows new interesting connections between strongly 2-nil-clean rings and weakly exchange rings
Determination of the biquaternion divisors of zero, including the idempotents and nilpotents
双四元数（复化四元数）代数包含幂等元（平方保持不变的元素）和幂零元（平方为零的元素）。它还包含零因子（具有消失范数的元素）。幂等元和幂零元是零的约数的子集。这些事实已经在文献中报道，但由于没有使用现代符号和术语收集在一起，因此仍然模糊不清。明确的公式找到所有的幂等元，幂零和零的除数似乎是不可用的文献中，我们纠正这与本文件。使用几种不同的双四元数表示，我们提出了简单的公式幂等，幂零和零的除数，我们表明，复分量的双四元数零的除数必须有一个平方和为零，这个条件是等价于两个条件的内积的实部和虚部的双四元数，以及实部和虚部的范数相等。我们给出了幂零、幂等和其他零因子的数值例子。最后，我们总结了一个关于零的双四元数因子集的组成，以及它的子集，幂等元和幂零元。
A ring $R$ is (strongly) 2-nil-clean if every element in $R$ is the sum of two idempotents and a nilpotent (that commute). Fundamental properties of such rings are discussed. Let $R$ be a 2-primal ring. If $R$ is strongly 2-nil-clean, we show that $M_n(R)$ is 2-nil-clean for all $n\in {\Bbb N}$. We also prove that the matrix ring is 2-nil-clean for a strongly 2-nil-clean ring of bounded index. These provide many classes of rings over which every matrix is the sum of two idempotents and a nilpotent.
Idempotent & Nilpotent Operators and Matrices in Bicomplex Space
本文研究双复空间中幂等和幂零算子的性质，定义相关矩阵并推导结果，提出若干定理以确定其存在性和行为条件。
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.
Products of Nilpotent and Idempotent Matrices over Finite Local Rings
本文研究有限交换局部主环R上的2阶矩阵，证明幂零与幂等矩阵的乘积必为幂等或幂零矩阵的乘积，IN-与NI-矩阵类重合，还给出其元素数量的显式公式。
Finite group algebras of nilpotent groups: a complete set of orthogonal primitive idempotents
给出了幂零群上有限群代数的正交本原幂等元完备集的一个显式构造。进一步，我们给出了幂零群的有限群代数的每个单满射像中的矩阵单位的完备集。
In this paper, we first study derivations in non nilpotent Lie triple algebras. We determine the structure of derivation algebra according to whether the algebra admits an idempotent or a pseudo-idempotent. We study the multiplicative structure of non nilpotent dimensionally nilpotent Lie triple algebras. We show that when $n=2p+1$ the adapted basis coincides with the canonical basis of the gametic algebra $G(2p+2,2)$ or this one obviously associated to a pseudo-idempotent and if $n=2p$ then the algebra is either one of the precedent case or a conservative Bernstein algebra.
{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.
Let $K$ be a field of characteristic zero, $\mathcal A$ a $K$-algebra and $δ$ a $K$-derivation of $\mathcal A$ or $K$-$\mathcal E$-derivation of $\mathcal A$ (i.e., $δ=\operatorname{Id}_A-ϕ$ for some $K$-algebra endomorphism $ϕ$ of $\mathcal A$). Motivated by the Idempotent conjecture proposed in [Z4], we first show that for every idempotent $e$ lying in both the kernel ${\mathcal A}^δ$ and the image $\operatorname{Im}δ\!:=δ({\mathcal A})$ of $δ$, the principal ideal $(e)\subseteq \operatorname{Im} δ$ if $δ$ is a locally finite $K$-derivation or a locally nilpotent $K$-$\mathcal E$-derivation …
A ring R is a Zhou nil-clean ring if every element in R is the sum of two tripotents and a nilpotent that commute. In this paper, Zhou nil-clean rings are further discussed with an emphasis on their relations with polynomials, idempotents and 2- idempotents. any a \in R, there exists e 2 Z[a] such that a-e \in R is nilpotent and e^5 = 5e^3 -4e, if and only if for any a 2 R, there exist idempotents e; f; g; h 2 Z[a] and a nilpotent w such that a = e+f +g+h+w, if and only if for any a 2 R, there exist 2-idempotents e; f 2 Z[a] and a nilpotent w 2 R such that a = e + f + w, if and only if for any…
We prove that if an involution in a ring is the sum of an idempotent and a nilpotent then the idempotent in this decomposition must be 1. As a consequence, we completely characterize weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl., DOI: 10.1142/S0219498816501486].
Rational group algebras of finite groups: from idempotents to units of integral group rings
给出了有限幂零群的有理群代数的正交本原幂等元完备集的一个无特征标的显式构造，并给出了这类代数的Wedderburn分解的完整描述.一个直接的后果是一个著名的结果Roquette的舒尔指数的简单组件的群代数有限幂零群群。作为应用，我们得到了有限幂零群$G$的整群环$\Z G$的单位群有一个由三个幂零群生成的有限指数子群，并给出了它们的生成元的明确描述.另一个应用是单位群中自由子群的新构造。在所有的结构处理，对子群$(H,K)$，称为强Shoda对，和明确的建设中心元素$e(G,H,K)$起着至关重要的作用。对任意有限群，证明了有理群代数的本原中心幂等元是这样的$e(G,H,K)$的有理线性组合，且$G$的子群中有$(H,K)$强Shoda偶.
Idempotents in nonassociative algebras and eigenvectors of quadratic operators
令$F$为一个字段，char $(F)\neq 2$。则每个有限维$F$ -代数有幂等元或绝对幂零元当且仅当在$F$上每个奇次多项式在$F$中有根。这也是$F$上有限维空间中所有二次算子存在特征向量的充要条件。
We introduce the concept of a weak nil clean ring, a generalization of nil clean ring, which is nothing but a ring with unity in which every element can be expressed as sum or difference of a nilpotent and an idempotent. Further if the idempotent and nilpotent commute the ring is called weak* nil clean. We characterize all $n\in \mathbb{N}$, for which $\mathbb{Z}_n$ is weak nil clean but not nil clean. We show that if $R$ is a weak* nil clean and $e$ is an idempotent in $R$, then the corner ring $eRe$ is also weak* nil clean. Also we discuss $S$-weak nil clean rings and their properties, where…
In the present paper we describe absolute nilpotent and some idempotent elements of an n- dimensional evolution algebra corresponding to two permutations and we decompose such algebras to the direct sum of evolution algebras corresponding to cycles of the permutations.
A partition of finite rings makes lifting possible
本文证明了每个有限环可按幂等元进行划分，并利用该划分实现了幂等元、幂零元、幂幺元、单位根及正则元等多种特殊元素的提升。
Matrix nil-clean factorizations over abelian rings
环$R$是零干净的，如果$R$中的每个元素都是幂等元与幂零元之和。环$R$是阿贝尔环，如果每个幂等元都是中心环.证明了若$R$是阿贝尔的，则$M_n(R)$是nil-clean的当且仅当$R/J(R)$是布尔的且$M_n(J(R))$是nil.这推广了Breaz等人的~ \cite{BGDT}和Kobalan等人的~ \cite{KLZ}。
Evolution algebra of a "chicken" population
我们考虑一个进化代数，它对应于一个两性群体，一组女性划分成许多不同的类型和男性只有一个类型。我们研究了代数的基本性质。这个代数是可交换的（因此是灵活的），一般来说不是结合的，也不一定是幂结合的。此外，它不是单一的。给出了代数的结构常数满足Jacobi恒等式和Jordan恒等式的一个条件，在该条件下，代数是结合的，交错的，幂结合的，幂零的.在一般情况下，我们描述了幂等元的全集合和绝对幂零元的全集合。描述了左（右）乘的所有算子的集合。在一定的结构常数条件下，证明了相应的代数是质心代数。并给出了二维代数和某些三维代数的分类。
Rational group algebras of generalized strongly monomial groups: primitive idempotents and units
本文针对有限广义强单项群，给出了有理群代数中 Schur 指数为 1 的简单分量上本原幂等元的显式计算方法，并描述了整群环单位群中由三个幂零群生成的有限指数子群，同时以奇阶 Frobenius 群为例说明理论的适用性。
Reflexivity of Rings via Nilpotent Elements
本文通过幂零元定义左 N-反射环，研究其性质，探讨其成为反射环和约化环的充分条件，并证明 N-反射性弱于反射性强于幂等反射性。
Nilpotency and Frattini theory for transposed Poisson algebras
研究转置泊松代数的幂零性和弗拉托尼理论，证明有限维代数的幂零性条件及弗拉托尼子代数性质，探讨其在不同结构下的分解与理想特性。
Classification of three-dimensional anti-dendriform algebras
本文分类了与结合性相关的反树形代数，重点处理零幂零结合代数及三维代数情形，并证明了含非零幂等元的结合代数不存在相容的反树形代数结构。
Strongly Nil-*-Clean Rings
一个 *-环$R$称为强nil-*-clean环，如果$R$的每个元素都是一个投影与一个幂零元之和，且这两个元素是可交换的.本文证明了$R$是强nil-*-clean环当且仅当$R$中的每一个幂等元都是投影，$R$是周期的，$R/J(R)$是布尔的.对任意交换 *-环$R$，证明了代数扩张$R[i]$其中$i^2=μi+η$对某些$μ,η\in R$是强nil-*-clean当且仅当$R$是强nil-*-clean且$μη$是幂零的.并给出了布尔 *-环与强nil-*-clean环之间的关系。
Few remarks on evolution algebras
本文研究了发展代数的一些代数性质。此外，我们将置换演化代数的研究归结为两种特殊类型的演化代数：幂等元和绝对幂零元。研究了三维发展代数，其发展基的每个元素都具有无限周期。此外，对于具有某些性质的演化代数，我们还刻画了它的结合包络代数。
Zero-product balanced algebras
该研究提出零积平衡代数的概念，证明其相关结构性质，刻画半素交换零积平衡代数的生成元特征，并得出有单位元的零积平衡代数的二分性结论。
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.
Rings Whose Non-Invertible Elements Are Nil-Clean
本研究系统探讨非可逆元可表为幂等元加幂零元的环类，刻画其关键性质并推导结构结果，延续了该领域两篇经典文献的研究方向。
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.
Evolution algebras: a survey of results and a list of open problems
本文综述演化代数的理论成果，列出50个开放问题与11个新猜想，还在证明助手Lean中验证了若干新结论，解决了部分猜想与问题。
On the monoid of partial order-preserving transformations of a finite chain whose domains and ranges are intervals
本文研究有限链上定义域和值域为区间的保序部分变换幺半群及其序递减子幺半群，给出其表示，并描述正则元素，确定秩、基数及幂等元和幂零元数量。
Natural generalized inverse and core of an element in semigroups, rings and Banach and Operator Algebras
利用最近的逆的概念沿一个元素在一个半群，和自然偏序的幂等元，我们研究双交换广义逆，并定义了一个新的逆称为自然逆，广义Drazin逆在一个半群，但也Koliha-Drazin逆在一个环。在这种情况下，我们得到一个类似于幂零分解，Kato或Mbekhta分解的核心分解。在Banach和算子代数中，我们指出了谱的研究是不充分的，并利用局部谱理论的思想来研究这种新的逆。
A discussion of bisexual populations with Wolbachia infection as an evolution algebra
本文用演化代数建模具有固定 CI 率与母系传播率的 Wolbachia 感染两性二倍体种群，刻画其代数结构、双权条件、幂等元及绝对幂零元性质。
Matsuo algebras in characteristic 2
本文将3-置换群相关的Matsuo代数理论扩展至特征2，改用与线关联的幂零元生成代数，给出了产生Z/2Z-分次融合律的完整分类，并在Sym(4)特例下揭示了更强的融合律及代数Miyamoto群结构。
$e$-Reduced rings in terms of the Zhou radical
本文利用Zhou根定义了环的Zhou右（左）$e$-约化性质，研究了其基本性质，给出若干例子来源，并探讨了相关扩张及全矩阵环的子环。
Absolute algebras, contramodules, and duality squares
本文提出绝对代数这一新型代数结构，通过对偶方块建立其与通常代数的同伦理论联系，证明线性对偶函子为Quillen函子并诱导同调有限性对象间的等价，进而将幂零李代数的同构判定定理推广到绝对李代数与绝对$\mathcal{L}_\infty$-代数。
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