Journal of the
Korean Mathematical Society JKMS

In this article we first investigate a sort of unit-IFP ring by which Antoine provides very useful information to ring theory in relation with the structure of coefficients of zero-dividing polynomials. Here we are concerned with the whole shape of units and nilpotent elements in such rings. Next we study the properties of unit-IFP rings through group actions of units on nonzero nilpotent elements. We prove that if $R$ is a unit-IFP ring such that there are finite number of orbits under the left (resp., right) action of units on nonzero nilpotent elements, then $R$ satisfies the descending chain condition for nil left (resp., right) ideals of $R$ and the upper nilradical of $R$ is nilpotent.

Keywords: unit-IFP ring, unit, nilpotent element, group action of units on nilpotent elements, descending chain condition for nil left ideals, orbit, nilradical, K\"othe's conjecture

MSC numbers: 16U60, 16P70, 16U80, 16N40