Abstract:
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 $S$ is a set of idempotents and show that if $S=\{0, 1\}$, then a $S$-weak nil clean ring contains a unique maximal ideal. Finally we show that weak* nil clean rings are exchange rings and strongly nil clean rings provided $2\in R$ is nilpotent in the later case. We have ended the paper with introduction of weak J-clean rings.

Abstract:
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 $i^2=\mu i+\eta$ for some $\mu,\eta\in R$ is strongly nil-*-clean if and only if $R$ is strongly nil-*-clean and $\mu\eta$ is nilpotent. The relationships between Boolean *-rings and strongly nil-*-clean rings are also obtained.

Abstract:
We characterize the nil clean matrix rings over fields. As a by product, it is proved that the full matrix rings with coefficients in commutative nil-clean rings are nil-clean, and we obtain a complete characterization of the finite rank Abelian groups with nil clean endomorphism ring and the Abelian groups with strongly nil clean endomorphism ring, respectively.

Abstract:
It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil rings, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some new results on graded-nil rings are also obtained.

We introduce nil 3-Armendariz rings, which are generalization of 3-Armendariz rings and nil Armendaiz rings and investigate their properties. We show that a ring R is nil 3-Armendariz ring if and only if for any , T_{n}(R) is nil 3-Armendariz ring. Also we prove that a right Ore ring R is nil 3-Armendariz if and only if so isQ, where Q is the classical right quotient ring of R. With the help of this result, we can show that a commutative ring R is nil 3-Armendariz if and only if the total quotient ring of R is nil 3-Armendariz.

Abstract:
The notion of clean rings and 2-good rings have many variations, and have been widely studied. We provide a few results about two new variations of these concepts and discuss the theory that ties these variations to objects and properties of interest to noncommutative algebraists. A ring is called nil-good if each element in the ring is the sum of a nilpotent element and either a unit or zero. We establish that the ring of endomorphisms of a module over a division is nil-good, as well as some basic consequences. We then define a new property we call nil-good clean, the condition that an element of a ring is the sum of a nilpotent, an idempotent, and a unit. We explore the interplay between these properties and the notion of clean rings.

Abstract:
本文研究具有对称自同态和对称导子的环. 利用性质nil(R[x]) =nil(R)[x], 我们证明了: 如果R是弱2-primal 环, 则R 是弱对称(σ, δ)-环当且仅当R[x] 是弱对称(σ,δ) -环. 本文结论拓展了关于对称环和弱对称环的研究. In this paper, we study rings with symmetric endomorphisms and symmetric derivations. By using the property nil(R[x]) =nil(R)[x], we show that if R is weakly 2-primal, then R is a weak symmetric (σ,δ)-ring if and only if R[x] is a weak symmetric (σ,δ)-ring, which extend the research on symmetric rings and weak symmetric rings

Abstract:
Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I[x] for some ideals I of R.

Abstract:
摘要： 介绍了诣零*-clean环和唯一诣零*-clean环的概念, 研究了这些环的基本性质和扩张性质,并讨论了几类*-环的关系。 Abstract: The concepts of nil *-clean rings and uniquely nil *-clean rings are introduced. Basic properties and extension properties of such rings are investigated. Moreover, the relations of several classes of *-rings are discussed

Abstract:
In this paper we discuss several constructions that lead to new examples of nil-clean, clean, and exchange rings. A characterization of the idempotents in the algebra defined by a 2-cocycle is given and used to prove some of the algebra's properties (the infinitesimal deformation case). From infinitesimal deformations we go to full deformations and prove that any formal deformation of a clean (exchange) ring is itself clean (exchange). Examples of nil-clean, clean, and exchange rings arising from poset algebras are also discussed.