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Korean Mathematical Society JKMS

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J. Korean Math. Soc. 2009; 46(5): 1027-1040

Printed September 1, 2009

https://doi.org/10.4134/JKMS.2009.46.5.1027

Copyright © The Korean Mathematical Society.

On conditions provided by Nilradicals

Hong Kee Kim, Nam Kyun Kim, Mun Seob Jeong, Yang Lee, Sung Ju Ryu, and Dong Eun Yeo

Gyeongsang National University, Hanbat National University, Busan National University, Busan National University, Busan National University, and Busan National University

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A ring $R$ is called IFP, due to Bell, if $ab=0$ implies $aRb=0$ for $a, b\in R$. Huh et al. showed that the IFP condition is not preserved by polynomial ring extensions. In this note we concentrate on a generalized condition of the IFPness that can be lifted up to polynomial rings, introducing the concept of quasi-IFP rings. The structure of quasi-IFP rings will be studied, characterizing quasi-IFP rings via minimal strongly prime ideals. The connections between quasi-IFP rings and related concepts are also observed in various situations, constructing necessary examples in the process. The structure of minimal noncommutative (quasi-)IFP rings is also observed.

Keywords: IFP ring, quasi-IFP ring, Wedderburn radical, nilradical, polynomial ring

MSC numbers: 16D25, 16N40, 16S36

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