On Nilpotent-invariant One-sided Ideals

The notion of a nilpotent-invariant module was introduced and thoroughly investigated in Koşan and Quynh (Comm. Algebra 45, 2775–2782 2017) as a proper extension of an automorphism-invariant module. In this paper a ring is called a right \(\mathfrak {n}\)-ring if every right ideal is nilpotent-invariant. We show that a right \(\mathfrak {n}\)-ring is the direct sum of a square full semisimple artinian ring and a right square-free ring. Moreover, right \(\mathfrak {n}\)-rings are shown to be stably finite, and if the ring is also an exchange ring then it satisfies the substitution property, has stable range 1. These results are non-trivial extensions of similar ones on rings every right ideal is automorphism-invariant.