[Paper Review] Weak Nil Clean Rings

Dhiren Kumar Basnet, Jayanta Bhattacharyya|arXiv (Cornell University)|Oct 26, 2015

Rings, Modules, and Algebras1 references3 citations

TL;DR

This paper introduces weak nil clean rings—rings where every element is expressible as the sum or difference of a nilpotent and an idempotent—generalizing nil clean and weakly clean rings. It characterizes all $\mathbb{Z}_n$ that are weak nil clean but not nil clean, proves that weak* nil clean rings are exchange rings and strongly nil clean when 2 is nilpotent, and introduces weak J-clean rings as a generalization of J-clean rings, establishing foundational lifting and decomposition properties.

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.

Motivation & Objective

• To generalize nil clean and weakly clean rings by introducing weak nil clean rings, where elements are sums or differences of nilpotents and idempotents.
• To characterize all $n \in \mathbb{N}$ such that $\mathbb{Z}_n$ is weak nil clean but not nil clean.
• To study $S$-weak nil clean rings for sets $S$ of idempotents, particularly when $S = \{0,1\}$, and prove such rings have a unique maximal ideal.
• To establish connections between weak* nil clean rings and exchange rings, and to show that if 2 is nilpotent, weak* nil clean rings are strongly nil clean.
• To introduce and initiate the study of weak J-clean rings as a generalization of J-clean rings, motivated by Problem 5 in [3].

Proposed method

• Define weak nil clean rings as rings in which every element is expressible as $n \pm e$ for $n$ nilpotent and $e$ idempotent.
• Introduce weak* nil clean rings by requiring that the nilpotent and idempotent elements commute.
• Use homomorphic image and direct product constructions to analyze closure properties and structural behavior.
• Prove that if $R$ is weak* nil clean and $2 \in \operatorname{Nil}(R)$, then $R$ is strongly nil clean, using module decomposition and endomorphism lifting.
• Define $S$-weak nil clean rings for subsets $S \subseteq \operatorname{Idem}(R)$, and show that when $S = \{0,1\}$, such rings have a unique maximal ideal.
• Introduce weak J-clean rings via decomposition $a = j \pm e$ with $j \in J(R)$, $e$ idempotent, and prove lifting and corner ring closure properties.

Experimental results

Research questions

• RQ1 For which natural numbers $n$ is $\mathbb{Z}_n$ weak nil clean but not nil clean?
• RQ2 What structural properties do $S$-weak nil clean rings possess, especially when $S = \{0,1\}$?
• RQ3 Under what conditions does a weak* nil clean ring become strongly nil clean?
• RQ4 How do weak* nil clean rings relate to exchange rings and strongly nil clean rings?
• RQ5 What are the lifting and corner ring properties of weak J-clean rings?

Key findings

• The paper characterizes all $n \in \mathbb{N}$ such that $\mathbb{Z}_n$ is weak nil clean but not nil clean, providing a complete classification.
• If $R$ is weak* nil clean and $2 \in \operatorname{Nil}(R)$, then $R$ is strongly nil clean, establishing a key link between weak* nil clean and strongly nil clean rings.
• For a weak* nil clean ring $R$, the corner ring $eRe$ is also weak* nil clean for any idempotent $e \in R$, preserving the property under corner ring restriction.
• When $S = \{0,1\}$, an $S$-weak nil clean ring has a unique maximal ideal, generalizing a result from commutative weakly nil clean rings.
• Weak* nil clean rings are exchange rings, and if $2 \in \operatorname{Nil}(R)$, they are also strongly nil clean, showing a strong structural constraint.
• The paper introduces weak J-clean rings and proves that if $R/J(R)$ is boolean and idempotents lift weakly modulo $J(R)$, then $R$ is weak J-clean, offering a new lifting condition.