On Ni and Quasi-Ni Rings

Abstract

Abstract. Let R be a ring. It is well-known that R is NI if and only if n i=0 Ra i R is a nil ideal of R whenever a polynomial n i=0 a i x i is nilpotent, where x is an indeterminate over R. We consider a condition which is similar to the preceding one: n i=0 Ra i R contains a nonzero nil ideal of R whenever n i=0 a i x i over R is nilpotent. A ring will be said to be 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.