The reflexive property for rings was introduced by Mason and play roles in noncommutative ring theory. A ring R is called reflexive if for a, b ∈ R, a Rb = 0 implies bRa = 0. Recently, Kheradmand et al. introduced the notion of RNP (reflexive-nilpotents-property) rings by restricting the reflexive property to nilpotent elements. In this article, we study reflexive-nilpotents-property skewed by a ring endomorphism α and introduce the notion of α-skew RNP rings. We investigate various properties and extensions of these rings and also determine the structure of minimal noncommutative α-skew RNP rings. Mathematics Subject Classification 16U99 • 16W20 • 16N40 1 Introduction Throughout this article, all rings are associative with unity, unless explicitly mentioned and all ring endomorphisms are nonzero. Given a ring R, the polynomial ring with an indeterminate x over R is denoted by R[x], the n by n full (resp., upper triangular) matrix ring over R by M n (R) (resp., U n (R)), D n (R) denotes the subring (a i j) ∈ U n (R) : diagonal entries of (a i j) are equal of U n (R), V n (R) denotes the ring of all matrices (a i j) ∈ D n (R) such that a i j = a (i+1)(j+1) for i = 1,. .. , n − 2 and j = 2,. .. , n − 1, N * (R) and N (R), respectively, denote the upper nilradical (i.e., the sum of all nil ideals) and the set of all nilpotent elements of R. Use E i j to denote the matrix with (i, j)-entry 1 and other entries 0, and Z n denotes the ring of integers modulo n. Recall that a ring R is called reduced, if it has no nonzero nilpotent elements. Due to Mason [16], a right ideal I of a ring R is called reflexive if for a, b ∈ R, a Rb ⊆ I implies bRa ⊆ I and a ring R is called reflexive, if the zero ideal of R is reflexive. It is well known that reflexive rings are generalizations of both commutative and reduced rings. Recently, Kheradmand et al. [10] generalized the notion of reflexive rings to RNP rings. A ring R is called RNP (reflexive-nilpotents-property) [10], if for a, b ∈ N (R), a Rb = 0 implies bRa = 0. Generalized reduced rings were extended to ring endomorphisms by Krempa. Following [12], an endomorphism α of a ring R is called rigid if for a ∈ R, aα(a) = 0 implies a = 0 and a ring R is called α-rigid, if there exists a rigid endomorphism α of R. Kwak et al. [14] extended the notion of reflexive rings to ring endomorphisms. Due to [14], an endomorphism α of a ring R is called right (resp., left) skew reflexive if for a, b ∈ R, a Rb = 0 implies bRα(a) = 0 (resp., α(b)Ra = 0) and a ring R is called right (resp., left) α-skew reflexive, if there exists a right (resp., left) skew reflexive endomorphism α of R. A ring R is called α-skew reflexive [14], if it is both right and left α-skew reflexive. Note that α-rigid rings are right α-skew reflexive by [14, Theorem 2.6].

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Mason introduced the notion of reflexive property for rings which play roles in noncommutative ring theory. In this paper, we extend this property to rings with involution and investigate their properties. We provide many examples of these rings and also consider some extensions such as trivial extension, Dorroh extension, etc.

The concept of strongly central reversible rings has been introduced in this paper. It has been shown that the class of strongly central reversible rings properly contains the class of strongly reversible rings and is properly contained in the class of central reversible rings. Various properties of the above-mentioned rings have been investigated. The concept of strongly central semicommutative rings has also been introduced and its relationships with other rings have been studied. Finally an open question raised in [D.