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
2025-08
General Letters in Mathematics (GLM)
(Issue : 2)
(Volume : 14)
Right Central CNZ Property Skewed by Ring Endomorphisms
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.
2024-06
General Letters in Mathematics (GLM)
(Issue : 1)
(Volume : 14)
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.
2024-05
Journal of Mathematical and Computational Science
(Issue : 1)
(Volume : 12)
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.
2022-04
General Letters in Mathematics (GLM)
(Issue : 1)
(Volume : 12)
Extensions of Nil-Reversible Rings with an Endomorphism α
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
2022-03
The General Letters in Mathematics
(Issue : 2)
(Volume : 9)
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 M1R is right a¯ -skew RIP.
2021-02
General letter in Mathmatics
(Issue : 1)
(Volume : 9)
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.
2019-09
Communications of the Korean Mathematical Society,
(Issue : 4)
(Volume : 32)
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.
2017-09
the Zanco Journal of Pure and Applied
(Issue : 3)
(Volume : 29)
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.
2017-08
Communications in Algebra
(Issue : 10)
(Volume : 45)
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.
2017-05
Communications in Algebra
(Issue : 11)
(Volume : 45)
Zero commutativity of nilpotent elements skewed by ring endomorphisms,
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.
2017-04
Thesis
2017
Zero Commutativity of Nilpotent Elements and Reflexivity with Maximal Ideal Axes
Non commutative algebra
2025
2007
On Lower Bounds of t – Blocking Sets in PG(2, q) and the Existence of Minimal Blocking Sets of Sizes 16 and 17 in PG(2, 9) .
