Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2010; 47(4): 719-733

Printed July 1, 2010

https://doi.org/10.4134/JKMS.2010.47.4.719

Copyright © The Korean Mathematical Society.

A structure on coefficients of nilpotent polynomials

Young Cheol Jeon, Yang Lee, and Sung Ju Ryu

Korea Science Academy, Busan National University, Busan National University

Abstract

We observe a structure on the products of coefficients of nilpotent polynomials, introducing the concept of $n$-semi-Armendariz that is a generalization of Armendariz rings. We first obtain a classification of reduced rings, proving that a ring $R$ is reduced if and only if the $n$ by $n$ upper triangular matrix ring over $R$ is $n$-semi-Armendariz. It is shown that $n$-semi-Armendariz rings need not be ($n+1$)-semi-Armendariz and vice versa. We prove that a ring $R$ is $n$-semi-Armendariz if and only if so is the polynomial ring over $R$. We next study interesting properties and useful examples of $n$-semi-Armendariz rings, constructing various kinds of counterexamples in the process.

Keywords: $n$-semi-Armendariz ring, polynomial ring, reduced ring, Armendariz ring, triangular matrix ring

MSC numbers: 16N40, 16S36