The symmetric property plays an important role in non-commutative ring theory and module theory. In this paper, we study the symmetric property with one element of the ring ℜ̂ and two nilpotent elements of ℜ̂ skewed by ring endomorphism ճ on rings, introducing the concept of a right ճ-𝒩ℒ -symmetric ring and extend the concept of right ճ-𝒩ℒ -symmetric rings to modules by introducing another concept called the right ճ-𝒩ℒ -symmetric module which is a generalization of ճ-symmetric modules. According to this, we examine the characterization of a right ճ-𝒩ℒ -symmetric ring and a right ճ-𝒩ℒ -symmetric module and their related properties including ring and explore their connections to other classes of rings and modules. Furthermore, we investigate the concept of ճ-𝒩ℒ -symmetric on some ring extensions and localizations like ℜ̂[ͷ], ℜ̂[ͷ, ͷ −1], Dorroh extension, Jordan extension and module localizations like Ω −1ℳ̂ Ω −1ℜ

The concept of the reversible ring property concerning nilpotent elements was introduced by A.M. Abdul-Jabbar and C. A. Ahmed, who introduced the concept of commutativity of nilpotent elements at zero, termed as a CNZ ring, as an extension of reversible rings. In this paper, we extend the CNZ property through the influence of a central ring endomorphism α, introducing a new type of ring called a right α-skew central CNZ ring. This concept not only expands upon CNZ rings but also serves as a generalization of right α-skew central reversible rings. We explore various properties of these rings and delve into extensions of right α-skew central CNZ rings, along with examining several established results, which emerge as corollaries of our findings.

The object of this paper is to present the notion of right CNZ rings with involutions, or, in short, right ∗−CNZ rings which are a generalization of right ∗−reversible rings and an extended of CNZ property . A ring R with involution ∗ is called right ∗−CNZ if for any nilpotent elements x, y ∈ R, xy = 0 implies yx∗ = 0. Every right ∗−CNZ ring with unity involution is CNZ but the converse need not be true in general, even for the commutative rings. In this note, we discussed some properties right ∗−CNZ ring. After that we explored right ∗−CNZ property on the extensions and localizations of the ring R.

In this paper, the concept of e−CNZ rings is introduced as a generalization of symmetric rings and a particular case of e−reversible rings. Regarding the question of how idempotent elements affect CNZ property of rings. In this note, we show that e−CNZ is not left-right symmetric. We present examples of right e−CNZ rings that are not CNZ and basic properties of right e − CNZ are provided. Some subrings of matrix rings and some extensions of rings such as Jordan extension are investigated in terms of right e−CNZ.

The concept of an α − nil reversible ring is a generalization of α − reversible ring as well as an extension of nil reversible rings. We first consider basic properties of α − nil reversible rings. Then we investigate extensions of α − nil reversible, including trivial extension, Dorroh extension and Jordan extension

T.K. Kwak and Y. Lee called a ring R satisfy the commutativity of nilpotent elements at zero[1] if ab = 0 for a, b 2 N(R) implies ba = 0. For simplicity, a ring R is called CNZ if it satisfies the commutativity of nilpotent elements at zero. In this paper we study an extension of a CNZ ring with its endomorphism. An endomorphism α of a ring R is called strong right ( resp., left) CNZ if whenever aα(b) = 0(resp., α(a)b = 0 ) for a, b 2 N(R) ba = 0. A ring R is called strong right (resp., left) α-CNZ if there exists a strong right (resp., left) CNZ endomorphism α of R, and the ring R is called strong α- CNZ if R is both strong left and right α- CNZ. Characterization of strong α- CNZ rings and their related properties including extensions are investigated . In particular, it’s shown that a ring R is reduced if and only if U2(R) is a CNZ ring. Furthermore extensions of strong α- CNZ rings are studied. ©2019 All rights reserved.

Abstract: The notion of an a-skew reflexive idempotent ring has been introduced in this paper to extend the concept of skew reflexive idempotent ring and that of an a-rigid ring. First basic properties of a-skew reflexive idempotent rings have been considered, including some examples needed in the process. It has been prove that for a ring R with an endomorphism a and n  2, if R satisfies the condition “eR fRfR = 0 implies eR f = 0 ”and R is a right a-skew RIP ring, then Vn(R) is a right a¯ -skew RIP ring. Also it has proven that if R is an algebra over a field K and D the Dorroh extension of R by K; where a is an endomorphism of R with a(1) = 1; then R is a right a-skew RIP ring if and only if D is a right a¯ -skew RIP ring. It’s shown that if M is a multiplicative closed subset of a ring R consisting of central regular elements and a an automorphism of R, then R is right a-skew RIP if and only if M􀀀1R is right a¯ -skew RIP.

Abstract The concept of central RM rings is introduced in this paper as a generalization of RM rings. Since every RM ring is central RM we study the sufficient condition for a central RM ring to be a RM one. It is shown that every central reversible and hence every central symmetric is central RM ring, however converse implications are wrong. It is also proven that the polynomial ring R[x] is central RM ring if R is central RM and quasi-Armendariz. Also -RM ring has been studied with it is central.©2019 All rights reserved.

Cohn called a ring R is reversible if whenever ab=0, then ba=0 for a,b∈R. The reversible property is an important role in noncommutative ring theory‎. ‎Recently‎, ‎Abdul-Jabbar et al‎. ‎studied the reversible ring property on nilpotent elements‎, ‎introducing‎ the concept of commutativity of nilpotent elements at zero (simply‎, ‎a CNZ ring)‎. ‎In this paper‎, ‎we extend the CNZ property of a ring as follows‎: ‎Let R be a ring and α an endomorphism of R‎, ‎we say that R is right (resp.‎, ‎left) α-nil-shifting ring if whenever aα(b)=0 (resp.‎, ‎α(a)b=0) for nilpotents a,b in R‎, ‎bα(a)=0 (resp.‎, ‎α(b)a=0)‎. ‎The characterization of α-nil-shifting rings and their related properties are investigated‎.

The reversible property of rings was initially introduced by Habeb and plays a role in noncommutative ring theory. In this note we study the reversible ring property on nilpotent elements, introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring) as a generalization of reversible rings. We first find the CNZ property of 2 by 2 full matrix rings over fields, which provides a basis for studying the structure of CNZ rings. We next observe various kinds of CNZ rings including ordinary ring extensions.

Abstract In the present paper, skewed reflexivity with maximal ideal axes by ring endomorphisms has been studied, introducing the concept of an α-skew RM rings, where α is an endomorphism of a given ring to extend the concept of an RM ring and that of an α-rigid ring. First some basic properties of an α-skew RM ring including some examples needed in the process have been considered. Then the characterization of a right α-skew RM ring and their related properties including the trivial extension and Dorroh extension have been investigated . In particular it is shown that for an endomorphism α and n ≥ 2 of a ring R. If R satisfies the condition “aM bRbR = 0 implies aM b = 0 ”and R is a right α-skew RM ring, then Vn(R) is right α¯-skew RM ring. Also the concept of an α-skew RMI ring has been observed . First some basic properties of α-skew RMI rings have been considered. Next α-skew RMI property of some kind of polynomial rings has been investigated.

The reflexive property for ideals was introduced by Mason and has important roles in noncommutative ring theory. We in this note study rings with the reflexivity whose axis is given by maximal ideals (simply, an RM ring) which are a generalization of symmetric rings. It is first shown that the reflexivity of a ring and the RM ring property are independent of each other, noting that both of them are generalizations of ideal-symmetric rings. We connect RM rings with reflexive rings in various situations raised naturally in the procedure. As a generalization of RM rings, we also study the structure of the reflexivity with the maximal ideal axis on idempotents (simply, an RMI ring) and then investigate the structure of minimal non-Abelian RMI rings (with or without identity) up to isomorphism.

The reversible property is an important role in noncommutative ring theory. Recently, the study of the reversible ring property on nilpotent elements is established by Abdul-Jabbar et al., introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring) as a generalization of reversible rings. We here study this property skewed by a ring endomorphism α, and such ring is called a right α-skew CNZ ring which is an extension of CNZ rings as well as a generalization of right α-skew reversible rings, and then investigate the structure of right α-skew CNZ rings and their related properties. Consequently, several known results are obtained as corollaries of our results.