Ikeda-Nakayama rings
The starting point for this paper is a well known result of M. Ikeda and T. Nakayama that asserts that if \(R\) is a right self-injective ring and \(A\) and \(B\) are right ideals of \(R\), then \(l(A\cap B)=l(A)+l(B)\), where \(l(X)\) denotes the left annihilator of a subset \(X\) of \(R\). The authors single out this condition and call the rings satisfying it right Ikeda-Nakayama rings (right IN-rings for short). They show that if \(R\) is a right IN-ring, then for each pair \(A,B\) of right ideals of \(R\) that are complements of each other, there exists an idempotent \(e\in R\) such that \(A=eR\) and \(B=(1-e)R\). From this it follows that a right IN-ring is right quasi-continuous and that \(R\) is right self-injective if and only if the \(2\times 2\) matrix ring \(M_2(R)\) is a right IN-ring. By a result of \textit{C. R. Hajarnavis} and \textit{N. C. Norton} [J. Algebra 93, 253-266 (1985; Zbl 0595.16009)] the rings \(R\) such that every right or left ideal of \(R\) is an annihilator (called dual rings) are left and right IN-rings. Here, this result is completed by showing 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. Moreover, it is also shown that every left perfect left and right IN-ring is QF but a left and right Artinian right IN-ring need not be QF.
scientific article; zbMATH DE number 1448623
scientific article