# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

## Article

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Bull. Korean Math. Soc. 2017; 54(6): 2107-2117

Published online November 30, 2017 https://doi.org/10.4134/BKMS.b160774

## Quasi-commutativity related to powers

Hyun-Min Kim, Dan Li, Zhelin Piao

Pusan National University, Pusan National University, Pusan National University

### Abstract

We study the quasi-commutativity in relation with powers of coefficients of polynomials. In the procedure we introduce the concept of {\it $\pi$-quasi-commutative} ring as a generalization of quasi-commutative rings. We show first that every $\pi$-quasi-commutative ring is Abelian and that a locally finite Abelian ring is $\pi$-quasi-commutative. The role of these facts are essential to our study in this note. The structures of various sorts of $\pi$-quasi-commutative rings are investigated to answer the questions raised naturally in the process, in relation to the structure of Jacobson and nil radicals.

Keywords: $\pi$-quasi-commutative ring, center, quasi-commutative ring, idempotent, polynomial ring, matrix ring, Abelian ring, locally finite ring

MSC numbers: 16U70, 16U80, 16S36

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