Ikeda-Nakayama rings

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 t wo left annihilators. In this paper we show that if R is a right IN-ring an d A and B are right ideals of R that are complements of each other, there e xists an idempotent e in R such that A = eR and B = (1 - e)R. As a conseque nce we show that R is right selfinjective if and only if M-2(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. Fi nally, we prove that R is quasi-Frobenius if and only if R is a left perfec t, left and right IN-ring, extending work on both selfinjective rings and d ual rings. Several examples are provided to show that our results are non-t rivial extensions of the known results on the subject. (C) 2000 Academic Pr ess.