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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.

16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16S70 Extensions of associative rings by ideals
Full Text: DOI
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