Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc. 2012; 49(3): 589-599

Printed May 1, 2012

https://doi.org/10.4134/BKMS.2012.49.3.589

Copyright © The Korean Mathematical Society.

Strongly nil clean matrices over $R[x]/\big(x^2-1\big)$

Huanyin Chen

Hangzhou Normal University

Abstract

An element of a ring is called strongly nil clean provided that it can be written as the sum of an idempotent and a nilpotent element that commute. We characterize, in this article, the strongly nil cleanness of $2\times 2$ and $3\times 3$ matrices over $R[x]/\big(x^2-1\big)$ where $R$ is a commutative local ring with characteristic $2$. Matrix decompositions over fields are derived as special cases.

Keywords: strongly nil matrix, characteristic polynomial, local ring

MSC numbers: 15A23, 13H99, 13B25