The biquaternion (complexified quaternion) algebra contains idempotents (elements whose square remains unchanged) and nilpotents (elements whose square vanishes). It also contains divisors of zero (elements with vanishing norm). The idempotents and nilpotents are subsets of the divisors of zero. These facts have been reported in the literature, but remain obscure through not being gathered together using modern notation and terminology. Explicit formulae for finding all the idempotents, nilpotents and divisors of zero appear not to be available in the literature, and we rectify this with the present paper. Using several different representations for biquaternions, we present simple formulae for the idempotents, nilpotents and divisors of zero, and we show that the complex components of a biquaternion divisor of zero must have a sum of squares that vanishes, and that this condition is equivalent to two conditions on the inner product of the real and imaginary parts of the biquaternion, and the equality of the norms of the real and imaginary parts. We give numerical examples of nilpotents, idempotents and other divisors of zero. Finally, we conclude with a statement about the composition of the set of biquaternion divisors of zero, and its subsets, the idempotents and the nilpotents.
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.
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
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.
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.
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.
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.
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.
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 provide an explicit construction for a complete set of orthogonal primitive idempotents of finite group algebras over nilpotent groups. Furthermore, we give a complete set of matrix units in each simple epimorphic image of a finite group algebra of a nilpotent group.
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].
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…
We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An immediate consequence is a well-known result of Roquette on the Schur indices of the simple components of group algebras of finite nilpotent groups. As an application, we obtain that the unit group of the integral group ring $\Z G$ of a finite nilpotent group $G$ has a subgroup of finite index that is generated by three nilpotent groups for which we have an ex…
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 …
An element in a ring $R$ is called uniquely weakly nil-clean if every element in $R$ can be uniquely written as a sum or a difference of a nilpotent and an idempotent in the sense of very idempotents. The structure of the ring in which every zero-divisor is uniquely weakly nil-clean is completely determined. We prove that every zero-divisor in a ring $R$ is uniquely weakly nil-clean if and only if $R$ is a D-ring, or $R$ is abelian, periodic, and $R/J(R)$ is isomorphic to a field $F$, ${\Bbb Z}_{3}\oplus {\Bbb Z}_{3}$, ${\Bbb Z}_{3}\oplus B$ where $B$ is Boolean, or a Boolean ring. As a specif…
A ring $R$ is nil-clean if every element in $R$ is the sum of an idempotent and a nilpotent. A ring $R$ is abelian if every idempotent is central. We prove that if $R$ is abelian then $M_n(R)$ is nil-clean if and only if $R/J(R)$ is Boolean and $M_n(J(R))$ is nil. This extend the main results of Breaz et al. ~\cite{BGDT} and that of Koşan et al.~\cite{KLZ}.
A *-ring $R$ is called a strongly nil-*-clean ring if every element of $R$ is the sum of a projection and a nilpotent element that commute with each other. In this article, we show that $R$ is a strongly nil-*-clean ring if and only if every idempotent in $R$ is a projection, $R$ is periodic, and $R/J(R)$ is Boolean. For any commutative *-ring $R$, we prove that the algebraic extension $R[i]$ where $i^2=μi+η$ for some $μ,η\in R$ is strongly nil-*-clean if and only if $R$ is strongly nil-*-clean and $μη$ is nilpotent. The relationships between Boolean *-rings and strongly nil-*-clean rings are …
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.
