Ikeda–Nakayama Rings

By V. Camillo, W. K. Nicholson, Mohamed Yousif. 2000

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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 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. 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.

Subfields: Algebra and Number Theory

View Paper: https://www.semanticscholar.org/paper/2ebf1864a371f4a916b91bcb379ea8627adbbeb7

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