• Title/Summary/Keyword: Near-rings

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A Characterization on Strong Reducibility of Near-Rings

  • Cho, Yong-Uk
    • 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.

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PRIMENESS AND PRIMITIVITY IN NEAR-RINGS

  • Wendt, Gerhard
    • 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.

A STUDY ON BAER AND P.P. NEAR-RINGS

  • Cho, Yong-Uk
    • East Asian mathematical journal
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    • v.16 no.1
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    • pp.105-110
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    • 2000
  • Baer rings were introduced by Kaplansky [3] to abstract various properties of rings of operators on a Hilbert spaces. On the other hand, p.p. rings were introduced by A. Hattori [2] to study the torsion theory. In this paper we introduce Baer near-rings and p.p. near-rings and study some properties and give some examples.

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LATTICE ORDERED SOFT NEAR RINGS

  • Mahmood, Tahir;Rehman, Zia Ur;Sezgin, Aslihan
    • 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.

NEAR-RINGS WITH LEFT BAER LIKE CONDITIONS

  • Cho, Yong-Uk
    • Bulletin of the Korean Mathematical Society
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    • v.45 no.2
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    • pp.263-267
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    • 2008
  • Kaplansky introduced the Baer rings as rings in which every left (or right) annihilator of each subset is generated by an idempotent. On the other hand, Hattori introduced the left (resp. right) P.P. rings as rings in which every principal left (resp. right) ideal is projective. The purpose of this paper is to introduce the near-rings with Baer like condition and near-rings with P.P. like condition which are somewhat different from ring case, and to extend the results of Arendariz and Jondrup.

A SPECIAL REDUCEDNESS IN NEAR-RINGS

  • Cho, Yong-Uk
    • 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.

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A NOTE ON STRONG REDUCEDNESS IN NEAR-RINGS

  • Cho, Yong-Uk
    • 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.

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ON STRONG FORM OF REDUCEDNESS

  • Cho, Yong-Uk
    • Honam Mathematical Journal
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    • v.30 no.1
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    • pp.1-7
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    • 2008
  • A near-ring N is said to be strongly reduced if, for a ${\in}$ N, $a^2{\in}N_c$ implies $a{\in}N_c$, where $N_c$ denotes the constant part of N. We investigate some properties of strongly reduced near-rings and apply those to the study of left strongly regular near-rings. Finally we classify all reduced and strongly reduced near-rings of order ${\leq}$ 7 using the description given in J. R. Clay [1].

A STUDY ON STRONGLY REDUCED AND REGULAR NEAR-RINGS

  • Cho, Yong-Uk
    • Proceedings of the Korean Institute of Intelligent Systems Conference
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    • 2008.04a
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    • pp.125-126
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    • 2008
  • A near-ring N is called strongly reduced if, for a ${\epsilon}$ N, $a^2\;{\epsilon}\;N_c$ implies a ${\epsilon}\;N_c$, where $N_c$ denotes the constant part of N. We investigate some properties of strongly reduced near-rings and apply those to the study of left strongly regular near-rings. Finally we classify some reduced, and strongly reduced near-rings.

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A STUDY ON WEAK BI-IDEALS OF NEAR-RINGS

  • Cho, Yong-Uk
    • East Asian mathematical journal
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    • v.24 no.2
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    • pp.145-149
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    • 2008
  • From the notion of bi-ideals in near-rings, various generalizations of regularity conditions have been studied. In this paper, we generalize further the notion of bi-ideals and introduce the notion of weak bi-ideals in near-rings and obtain some characterizations using this concept in left self distributive near-rings.

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