Korean J. Math.  Vol 24, No 3 (2016)  pp.307-317
DOI: https://doi.org/10.11568/kjm.2016.24.3.307

On NI and quasi-NI rings

Dong Hwa Kim, Seung Ick Lee, Yang Lee, Sang Jo Yun

Abstract


Let $R$ be a ring. It is well-known that $R$ is {\it NI} if and only if $\sum_{i=0}^nRa_iR$ is a nil ideal of $R$ whenever a polynomial $\sum_{i=0}^na_ix^i$ is nilpotent, where $x$ is an indeterminate over $R$. We consider a condition which is similar to the preceding one:
$\sum_{i=0}^nRa_iR$ contains a nonzero nil ideal of $R$ whenever $\sum_{i=0}^na_ix^i$ over $R$ is nilpotent. A ring will be said to be {\it quasi-NI} if it satisfies this condition. The structure of quasi-NI rings is observed, and various examples are given to situations which raised naturally in the process.


Keywords


quasi-NI ring, NI ring, polynomial ring, matrix ring

Subject classification

16D25, 16N40, 16S36

Sponsor(s)

This work was supported by 2-year Research Grant of Pusan National University.

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