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A study on quasi-duo rings. (English) Zbl 0938.16002
A ring \(R\) (associative with identity) is called right (left) duo if every right (left) ideal of \(R\) is two-sided. A ring \(R\) is called right (left) quasi-duo if every right (left) maximal ideal of \(R\) is two-sided. A ring \(R\) is called weakly right (left) duo if for each \(a\in R\) there exists a positive integer \(n\), depending on \(a\), such that \(a^nR\) (\(Ra^n\)) is a two-sided ideal of \(R\). As another generalization of commutative rings, there are 2-primal rings. A ring \(R\) is called 2-primal if the prime radical of \(R\) is equal to the set of all nilpotent elements in \(R\). In this paper the authors present some examples to give negative answers to the following questions (they also give positive answers in very special cases): 1) Are right quasi-duo rings 2-primal? 2) Are formal power series rings over weakly right duo rings also weakly right duo? 3) Are 2-primal rings right quasi-duo?

16D25 Ideals in associative algebras
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16U80 Generalizations of commutativity (associative rings and algebras)