Structure of NI rings related to centers

Abstract

We first obtain that NI rings satisfy a property that if ab is central for elements a, b, then ${(ab)}^n={(ba)}^n$ for some $n\ge 1$, by applying a property of reduced rings. We prove next the following: Let R be a ring and I be the ideal of R generated by the subset $\{ab-ba|a,b\in R\mathrm{ }\text{such}\mathrm{ }\text{that}ab\mathrm{ }\text{is}\mathrm{ }\text{central}\mathrm{ }\text{in}\mathrm{ }R\}$. (i) Suppose that ab is central for $a,b\in R$ and ab – ba is a nonzero nilpotent. Then, $A(ab-ba)A$ is a nonzero nilpotent ideal of the subring A of R, where 1 is the identity of R, $B=\mathbb{Z}·1=\left\{n1\right|n\in \mathbb{Z}\}$, and A is the algebra $B〈a,b〉$ generated by a, b over B. (ii) If R is NI, then I is nil and R/I is an Abelian NI ring. (iii) Let R be reversible and ab be central for $a,b\in R$. Then, there exists $l\ge 1$ such that, for every $n\ge l,{(ab)}^n={(ba)}^n$ and ${(ab)}^n=b^h{(ab)}^{n-h}{a}^h$ for all $1\le h\le n$; especially $a^n{b}^n={(ab)}^n=b^n{a}^n$. We call a ring pseudo-NI if it satisfies the first property of NI rings to be mentioned and examine the structures of NI and pseudo-NI rings in several ring theoretic situations, showing that semisimple Artinian rings are pseudo-NI.

Keywords:

• Abelian ring
• center
• NI ring
• nilpotent
• nilradical
• pseudo-NI ring
• reduced ring
• reversible ring
• semisimple Artinian ring
• upper triangular matrix ring

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 16U70
• 16N40
• 16S50

Acknowledgments

The authors thank the referee for a very careful reading of the manuscript and many valuable suggestions that greatly improved the paper.

Additional information

Funding

This study was supported by research funds from Dong-A University.