Ikeda–Nakayama Rings

Abstract

A ring R is called a right Ikeda-Nakayama ring (right IN-ring) if the left annihilator of the intersection of any two right ideals is the sum of the two left annihilators. In this paper we show that if R is a right IN-ring and A and B are right ideals of R that are complements of each other, there exists an idempotent e in R such that A=eR and B=(1−e)R. As a consequence we show that R is right selfinjective if and only if M2(R) is a right IN-ring. It is also shown that R is a dual ring if and only if R is a left and right IN-ring and the dual of every simple right R-module is simple. Finally, we prove that R is quasi-Frobenius if and only if R is a left perfect, left and right IN-ring, extending work on both selfinjective rings and dual rings. Several examples are provided to show that our results are non-trivial extensions of the known results on the subject.