Journal of the
Korean Mathematical Society
JKMS

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

Article

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J. Korean Math. Soc. 2019; 56(5): 1403-1418

Online first article July 17, 2019      Printed September 1, 2019

https://doi.org/10.4134/JKMS.j180698

Copyright © The Korean Mathematical Society.

Annihilating content in polynomial and power series rings

Emad Abuosba, Manal Ghanem

The University of Jordan; The University of Jordan

Abstract

Let $R$ be a commutative ring with unity. If $f(x)$ is a zero-divisor polynomial such that $f(x)=c_{f}f_{1}(x)$ with $c_{f}\in R$ and $f_{1}(x)$ is not zero-divisor, then $c_{f}$ is called an annihilating content for $ f(x) $. In this case $Ann(f)=Ann(c_{f})$. We defined EM-rings to be rings with every zero-divisor polynomial having annihilating content. We showed that the class of EM-rings includes integral domains, principal ideal rings, and PP-rings, while it is included in Armendariz rings, and rings having a.c. condition. Some properties of EM-rings are studied and the zero-divisor graphs $\Gamma (R)$ and $\Gamma (R[x])$ are related if $R$ was an EM-ring. Some properties of annihilating contents for polynomials are extended to formal power series rings.

Keywords: polynomial ring, power series ring, annihilating content, EM-ring, generalized morphic ring, zero-divisor graph

MSC numbers: 13F20, 13F25, 13E05

Supported by: The second author was supported by the “Scientific Research Deanship” at “The Univer-sity of Jordan”.