• Communications of Mathematical Education
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• v.10
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• pp.283-292
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• 2000

We shall introduce new concepts of near-rings, that is, strong reducibility and left semi ${\pi}$-regular near-rings. We will study every strong reducibility of near-ring implies reducibility of near-ring but this converse is not true, and also some characterizations of strong reducibility of near-rings. We shall investigate some relations between strongly reduced near-rings and left strongly regular near-rings, and apply strong reducibility of near-rings to the study of left semi ${\pi}$-regular near-rings, s-weekly regular near-rings and some other regularity of near-rings.

• Korean Journal of Mathematics
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• v.26 no.3
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• pp.503-517
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• 2018

Keeping in view the expediency of soft sets in algebraic structures and as a mathematical approach to vagueness, in this paper the concept of lattice ordered soft near rings is introduced. Different properties of lattice ordered soft near rings by using some operations of soft sets are investigated. The concept of idealistic soft near rings with respect to lattice ordered soft near ring homomorphisms is deliberated.

• Journal of the Korean Mathematical Society
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• v.58 no.2
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• pp.309-326
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• 2021

In near-ring theory several different types of primitivity exist. These all imply several different types of primeness. In case of near-rings with DCCN most of the types of primeness are known to imply primitivity of a certain kind. We are able to show that also so called 1-prime near-rings imply 1-primitivity. This enables us to classify maximal ideals in near-rings with chain condition with the concept of 1-primeness which leads to further results in the structure theory of near-rings.

• East Asian mathematical journal
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• v.22 no.1
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• pp.61-69
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• 2006

A near-ring N is reduced if, for $a{\in}N,\;a^2=0$ implies a=0, and N is left strongly regular if for all $a{\in}N$ there exists $x{\in}N$ such that $a=xa^2$. Mason introduced this notion and characterized left strongly regular zero-symmetric unital near-rings. Several authors ([2], [5], [7]) studied these properties in near-rings. Reddy and Murty extended some results in Mason to the non-zero symmetric case. In this paper, we will define a concept of strong reducedness and investigate a relation between strongly reduced near-rings and left strongly regular near-rings.

• East Asian mathematical journal
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• v.29 no.3
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• pp.255-258
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• 2013

In this paper, we initiate the study of zero symmetric and constant parts of near-rings, and then apply these to self map near-rings. Next, we investigate that every near-ring can be embedded into some self map near-ring, and every zero symmetric near-ring can be embedded into some zero symmetric self map near-ring.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.523-527
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• 2011

In this paper, We initiate a study of zero symmetric and constant parts of near-rings, and then apply these to self map near-rings. Next, we investigate that every near-ring can be embedded into some self map near-ring, and every zero symmetric near-ring can be embedded into some zero symmetric self map near-ring.

• Communications of the Korean Mathematical Society
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• v.28 no.4
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• pp.669-677
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• 2013

In this paper we introduce the notion of NI near-rings similar to the notion introduced in rings. We give topological properties of collection of strongly prime ideals in NI near-rings. We have shown that if N is a NI and weakly pm near-ring, then $Max(N)$ is a compact Hausdorff space. We have also shown that if N is a NI near-ring, then for every $a{\in}N$, $cl(D(a))=V(N^*(N)_a)=Supp(a)=SSpec(N){\setminus}int\;V(a)$.

• The Pure and Applied Mathematics
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• v.10 no.4
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• pp.199-206
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• 2003

Let N be a right near-ring. N is said to be strongly reduced if, for $a\inN$, $a^2 \in N_{c}$ implies $a\;\in\;N_{c}$, or equivalently, for $a\inN$ and any positive integer n, $a^{n} \in N_{c}$ implies $a\;\in\;N_{c}$, where $N_{c}$ denotes the constant part of N. We will show that strong reducedness is equivalent to condition (ⅱ) of Reddy and Murty's property $(^{\ast})$ (cf. [Reddy & Murty: On strongly regular near-rings. Proc. Edinburgh Math. Soc. (2) 27 (1984), no. 1, 61-64]), and that condition (ⅰ) of Reddy and Murty's property $(^{\ast})$ follows from strong reducedness. Also, we will investigate some characterizations of strongly reduced near-rings and their properties. Using strong reducedness, we characterize left strongly regular near-rings and ($P_{0}$)-near-rings.

• East Asian mathematical journal
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• v.25 no.4
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• pp.441-447
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• 2009

Jat and Choudhari defined a near-ring R with left bipotent or right bipotent condition in 1979. Also, we can dene a near-ring R as subcommutative if aR = Ra for all a in R. From these above two concepts it is natural to investigate the near-ring R with the properties aR = $Ra^2$ (resp. $a^2R$ = Ra) for each a in R. We will say that such is a near-ring with (1,2)-potent condition (resp. a near-ring with (2,1)-potent condition). Thus, we can extend a general concept of a near-ring R with (m,n)-potent condition, that is, $a^mR\;=\;Ra^n$ for each a in R, where m, n are positive integers. We will derive properties of near-ring with (1,n) and (n,1)-potent conditions where n is a positive integer, any homomorphic image of (m,n)-potent near-ring is also (m,n)-potent, and we will obtain some characterization of regular near-rings with (m,n)-potent conditions.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.3
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• pp.225-229
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• 2006

In this paper, we introduce some concepts of ${\Gamma}$-near-ring and obtain their properties on ${\Gamma}$-near-rings through regularity conditions.