As a generalization of nil clean ideal, we define weak nil clean ideal of a ring. An ideal $I$ of a ring $R$ is weak nil clean ideal if for any $x\in I$, either $x=e+n$ or $x=-e+n$, where $n$ is a nilpotent element and $e$ is an idempotent element of $R$. Some interesting properties of weak nil clean ideal and its relation with weak nil clean ring have been discussed.

Dhiren Kumar Basnet Ajay Sharma

Every quasi completely regular semiring is a b-lattice of completely Archimedean semirings, i.e., a b-lattice of nil-extensions of completely simple semirings. In this paper we consider the semiring which is a b-lattice of nil-extensions of orthodox completely simple semirings.

S. K. Maity R. Chatterjee

In this article, we study higher Nil $K$-groups via binary complexes. More particularly, we exhibit an explicit form of generators of higher Nil $K$-groups in terms of binary complexes.

Sourayan Banerjee Vivek Sadhu

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 Ko\c{s}an et al.~\cite{KLZ}.

Huanyin Chen

In this article, we have defined nil clean graph of a ring $R$. The vertex set is the ring $R$, two ring elements $a$ and $b$ are adjacent if and only if $a + b$ is nil clean in $R$. Graph theoretic properties like girth, dominating set, diameter etc. of nil clean graph have been studied for finite commutative rings.

Dhiren Kumar Basnet Jayanta Bhattacharyya

Motivated by the concept of clean ideals, we introduce the notion of nil clean ideals of a ring. We define an ideal $I$ of a ring $R$ to be nil clean ideal if every element of $I$ can be written as a sum of an idempotent and a nilpotent element of $R$. In this article we discuss various properties of nil clean ideals.

Ajay Sharma Dhiren Kumar Basnet

In this paper, we introduce a new concept in Nil-semicommutative modules and present it as an extension of Nil-semicommutative rings to modules. We prove that the class of Nil-semicommutative modules is contained in the class of Weakly semicommutative modules while that of the converse may not be true. We also show that in case of Semicommutative modules and Nil-semicommutative modules, one does not imply the other. Moreover, for a given Nil-semicommutative ring, we provide the conditions under which the same can be extended to a Nil-semicommutative module. Lastly, we also prove that for a left $R$-module $M$, $_RM$ is Nil-semicommutative iff it's localization $S^{-1}M$ over the ring $S^{-1}R$ is also Nil-semicommutative.Various other examples and propositions highlighting the comparative studies of this new class of modules with different classes of modules are also discussed in order to validate the concept.

M. Rhoades Kh. Herachandra Nazeer Ansari

We consider in-depth and characterize in certain aspects the class of so-called {\it strongly NUS-nil clean rings}, that are those rings whose non-units are {\it square nil-clean} in the sense that they are a sum of a nilpotent and a square-idempotent that commutes with each other. This class of rings lies properly between the classes of strongly nil-clean rings and strongly clean rings. In fact, it is proved the valuable criterion that a ring $R$ is strongly NUS-nil clean if, and only if, $a^4-a^2\in Nil(R)$ for every $a\not\in U(R)$. In particular, a ring $R$ with only trivial idempotents is strongly NUS-nil clean if, and only if, $R$ is a local ring with nil Jacobson radical. Some special matrix constructions and group ring extensions will provide us with new sources of examples of NUS-nil clean rings.

Mina Doostalizadeh Ahmad Moussavi Peter Danchev

We give a criterion for smoothness of a point in any Schubert variety in any G/B in terms of the nil Hecke ring.

Shrawan Kumar

We construct a nil algebra over a countable field which has finite but non-zero Gelfand-Kirillov dimension.

T H Lenagan Agata Smoktunowicz