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Reflexive property on nil ideals. (English) Zbl 1359.16016
In this paper, the authors introduce a new class of so called QTR-nil-reflexive rings, that is, associative rings $$R$$ with identity such that $$IJ=0$$ implies $$JI=0$$, for all nil ideals $$I,J$$ of $$R$$. They show that nil-reflexive rings form a proper generalization of reflexive rings, that is, rings $$A$$ such that $$aAb=0$$ implies $$bAa=0$$, for every $$a,b\in A$$. The authors give some characterizations of nil-reflexive rings, describe some of their properties and show how they are related to the famous Koethe’s problem which asks whether the sum of two nil left ideals of a ring is nil. Moreover, the authors study some extensions of nil-reflexive rings such as the Dorroh extension, matrix rings, polynomial rings, power series rings and the right quotient rings. For example, the authors show that, in general, nil-reflexivity does not pass to subrings, homomorphic images, polynomials nor power series rings and describe some cases when they do.

##### MSC:
 16N40 Nil and nilpotent radicals, sets, ideals, associative rings 16S70 Extensions of associative rings by ideals
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##### References:
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