• 충청수학회지
• /
• 제22권3호
• /
• pp.293-297
• /
• 2009

We will introduce some properties of strongly reduced near-rings and the notion of left $\pi$-regular near-ring. Also, we will study for right $\pi$-regular, strongly left $\pi$-regular, strongly right $\pi$-regular and strongly $\pi$- regular. Next, we may characterize the strongly $\pi$-regular near-rings with related strong reducibility.

• Journal of applied mathematics & informatics
• /
• 제28권1_2호
• /
• pp.509-513
• /
• 2010

In this paper, we will investigate some properties of strongly reduced near-rings. The purpose of this paper is to find more characterizations of the strong regularity in near-rings, which are closely related with strongly reduced near-rings.

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제17권2호
• /
• pp.131-136
• /
• 2010

Throught this paper, we will investigate some properties of left regular and strongly reduced near-rings. Mason introduced the notion of left regularity and he characterized left regular zero-symmetric unital near-rings. Also, this concept have been studied by several authors. The purpose of this paper is to find some characterizations of the strong reducibility in near-rings, and the strong regularity in near-rings which are closely related with strongly reduced near-rings.

• East Asian mathematical journal
• /
• 제22권1호
• /
• pp.61-69
• /
• 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.

• 대한수학회보
• /
• 제59권4호
• /
• pp.853-868
• /
• 2022

We study the structure of right regular commutators, and call a ring R strongly C-regular if ab - ba ∈ (ab - ba)²R for any a, b ∈ R. We first prove that a noncommutative strongly C-regular domain is a division algebra generated by all commutators; and that a ring (possibly without identity) is strongly C-regular if and only if it is Abelian C-regular (from which we infer that strong C-regularity is left-right symmetric). It is proved that for a strongly C-regular ring R, (i) if R/W(R) is commutative, then R is commutative; and (ii) every prime factor ring of R is either a commutative domain or a noncommutative division ring, where W(R) is the Wedderburn radical of R.

• 대한수학회지
• /
• 제52권4호
• /
• pp.839-851
• /
• 2015

A *-ring R is called (strongly) *-clean if every element of R is the sum of a projection and a unit (which commute with each other). In this note, some properties of *-clean rings are considered. In particular, a new class of *-clean rings which called strongly ${\pi}$-*-regular are introduced. It is shown that R is strongly ${\pi}$-*-regular if and only if R is ${\pi}$-regular and every idempotent of R is a projection if and only if R/J(R) is strongly regular with J(R) nil, and every idempotent of R/J(R) is lifted to a central projection of R. In addition, the stable range conditions of *-clean rings are discussed, and equivalent conditions among *-rings related to *-cleanness are obtained.

• Kyungpook Mathematical Journal
• /
• 제64권1호
• /
• pp.171-184
• /
• 2024

In this paper, we show that if G is strongly regular then the Gallai graph Γ(G) and the anti-Gallai graph ∆(G) of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well as conditions under which they are 2-connected. We include also a number of concrete examples and a discussion of spectral properties of the Gallai and anti-Gallai graphs.

• 대한수학회논문집
• /
• 제27권3호
• /
• pp.483-488
• /
• 2012

In this paper we introduce the notion of P-strongly regular near-ring. We have shown that a zero-symmetric near-ring N is P-strongly regular if and only if N is P-regular and P is a completely semiprime ideal. We have also shown that in a P-strongly regular near-ring N, the following holds: (i) $Na$ + P is an ideal of N for any $a{\in}N$. (ii) Every P-prime ideal of N containing P is maximal. (iii) Every ideal I of N fulfills I + P = $I^2$ + P.

• 대한수학회논문집
• /
• 제22권3호
• /
• pp.323-330
• /
• 2007

In this paper we introduce the notion of strongly regular near-subtraction semigroups (right). We have shown that a near-subtraction semigroup X is strongly regular if and only if it is regular and without non zero nilpotent elements. We have also shown that in a strongly regular near-subtraction semigroup X, the following holds: (i) Xa is an ideal for every a $\in$ X (ii) If P is a prime ideal of X, then there exists no proper k-ideal M such that P $\subset$ M (iii) Every ideal I of X fulfills $I=I^2$.

• Kyungpook Mathematical Journal
• /
• 제60권4호
• /
• pp.839-851
• /
• 2020

We define morphic near-ring elements and study their behavior in regular near-rings. We show that the class of left morphic regular near-rings is properly contained between the classes of left strongly regular and unit-regular near-rings.