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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?

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