1

Nil-clean and strongly nil-clean rings by Tamer Koşan, Zhou Wang, and Yiqiang Zhou.

Theorem 2.1. An element $a\in R$ is strongly nil-clean iff $a$ is strongly clean in $R$ and $a−a^2$ is a nilpotent.

Proof (⇒) . Let $a=e+b$ be a strongly nil-clean decomposition in $R$. Then $a=(1−e)+(2e−1+b)$ is a strongly clean decomposition in $R$.

Why does $2e-1+b$ is a unit?

Definitions

$a$ is said to be strongly nil-clean if $a=e+b$ where $e$ is idempotent and $b$ is nilpotent, and $eb=be$.

$a$ is said to be strongly clean if $a=e+b$ where $e$ is idempotent, $b$ is a unit, and $eb=be$.

rschwieb
  • 160,592
chandara
  • 95
  • Sorry, earlier I misread the intention because no definitions were included. When I investigated this morning, I saw the way it works and changed my solution. You should really include definitions when they are not commonly known, and this is the case for these variations of clean elements. – rschwieb Dec 02 '19 at 15:32

1 Answers1

0

By hypothesis, $a=e+b$ with $e$ idempotent and $b$ nilpotent, and $eb=be$.

$2e-1$ is a unit with inverse $2e-1$, and it commutes with the nilpotent element $b$. By a well-known lemma, their sum is a unit.

rschwieb
  • 160,592