Annihilating content in polynomial and power series rings
J. Korean Math. Soc. 2019 Vol. 56, No. 5, 1403-1418
https://doi.org/10.4134/JKMS.j180698
Published online September 1, 2019
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
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