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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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