Journal of the
Korean Mathematical Society JKMS

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 need not be preserved by polynomial ring extensions. But it is shown that $\sum_{i=0}^nEa_iE$ is a nonzero nilpotent ideal of $E$ whenever $R$ is an IFP ring and $0\neq f\in F$ is nilpotent, where $E$ is a polynomial ring over $R$, $F$ is a polynomial ring over $E$, and $a_i$'s are the coefficients of $f$. We shall use the term $near$-IFP to denote such a ring as having place near at the IFPness. In the present note the structures of IFP rings and near-IFP rings are observed, extending the classes of them. IFP rings are NI (i.e., nilpotent elements form an ideal). It is shown that the near-IFPness and the NIness are distinct each other, and the relations among them and related conditions are examined.

Keywords: IFP ring, near-IFP ring, reduced ring, NI ring, polynomial ring, matrix ring, nilpotent ideal

MSC numbers: 16D25, 16N40, 16N60