Abstract
In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce nil-reflexive rings. It is shown that the notion of nil-reflexive is a generalization of that of nil-semicommutativity. Examples are given to show that nil-reflexive rings need not be reflexive and vice versa, and nil-reflexive rings but not semicommutative are presented. We also proved that every ring with identity is weakly reflexive defined by Zhao, Zhu and Gu. Moreover, we investigate basic properties of nil-reflexive rings and provide some source of examples for this class of rings. We consider some extensions of nil-reflexive rings, such as trivial extensions, polynomial extensions and Nagata extensions.
Highlights
Throughout this paper all rings are associative with identity unless otherwise stated
We deal with a new approach to re‡exive property for rings by using nilpotent elements, in this direction we introduce nil-re‡exive rings
Mason introduced the re‡exive property for ideals, and this concept was generalized by some authors, de...ning idempotent re‡exive right ideals and rings, completely re‡exive rings, weakly re‡exive rings
Summary
Throughout this paper all rings are associative with identity unless otherwise stated. Reduced rings are completely re‡exive and every completely re‡exive ring is semicommutative, i.e. according to [12], a ring R is called semicommutative if for all a, b 2 R, ab = 0 implies aRb = 0. This is equivalent to the de...nition that any left (right) annihilator of R is an ideal of R. In this paper it is proved that the class of nil-re‡exive rings lies strictly between the classes of nil-semicommutative rings and weakly re‡exive rings. We write R[x], U (R), P (R), and Sn(R) (Vn(R)) for the polynomial ring over a ring R, the set of invertible elements, the prime radical of R, and the subring consisting of all upper triangular matrices over a ring R with equal main diagonal (every diagonal) entries, respectively
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