On nilpotent-invariant one-sided ideals (Q6563082)

In [Commun. Algebra 45, No. 7, 2775--2782 (2017; Zbl 1387.16003)], \textit{M. T. Koşan} and \textit{T. C. Quynh} introduced and studied the concept of nilpotent-invariant module. Recall that a module \(M\) is said to be \textit{nilpotent-invariant} if \(f(M)\leq M\) for all nilpotent endomorphisms of its injective envelope \(E(M)\). This is a proper extension of the automorphism-invariant module class, where by a module is said to be \textit{automorphism-invariant} if it is invariant under all automorphisms of its injective envelope. This class of modules is studied by \textit{T.-K. Lee} and \textit{Y. Zhou} [J. Algebra Appl. 12, No. 2, Paper No. 1250159, 9 p. (2013; Zbl 1263.16005)].\N\N\NIn this paper, a ring \(R\) is called a \textit{right \(\mathfrak{n}\)-ring} if every right ideal of \(R\) is nilpotent-invariant. The authors show that a right \(\mathfrak{n}\)-ring is the direct sum of a square full\Nsemisimple artinian ring and a right square-free ring. Moreover, right \(\mathfrak{n}\)-ring are shown to be stably finite, and if the ring is also an exchange ring then it satisfies the substitution property,\Nhas stable range 1. Note that, these results are non-trivial extensions of similar ones on rings every\Nright ideal is automorphism-invariant. Also, they prove that a regular right \(\mathfrak{n}\)-ring with zero right socle is strongly regular. Finally, the authors study nilpotent-invariant modules over formal triangular matrix rings. In particular, they give a necessary condition for formal triangular matrix rings to be right \(\mathfrak{n}\)-rings.