DOI: https://doi.org/10.11568/kjm.2016.24.3.307

### On NI and quasi-NI rings

#### 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

#### Subject classification

16D25, 16N40, 16S36#### Sponsor(s)

This work was supported by 2-year Research Grant of Pusan National University.#### Full Text:

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