Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc. 2011; 48(1): 115-127

Printed January 1, 2011

https://doi.org/10.4134/BKMS.2011.48.1.115

Copyright © The Korean Mathematical Society.

A concept unifying the Armendariz and NI conditions

Young Chun, Young Cheol Jeon, Sungkyung Kang, Key Nyoung Lee, and Yang Lee

Korea Science Academy, Korea Science Academy, Korea Science Academy, Korea Science Academy, Pusan National University

Abstract

We study the structure of the set of nilpotent elements in various kinds of ring and introduce the concept of NR ring as a generalization of Armendariz rings and $NI$ rings. We determine the precise relationships between $NR$ rings and related ring-theoretic conditions. The K\"othe's conjecture is true for the class of $NR$ rings. We examined whether several kinds of extensions preserve the $NR$ condition. The classical right quotient ring of an $NR$ ring is also studied under some conditions on the subset of nilpotent elements.

Keywords: $NR$ ring, $NI$ ring, Armendariz ring, matrix ring

MSC numbers: 16D25, 16N40