Morphic Elements in Regular Near-rings

2021

Some results on r-regular (r-RN) and also in s-weakly regular (s-WRN)near-rings were established in this article. It is proved that for a near-ring H 0  is s-WRN, then H is simple iff H is integral. And also proved that for an r-RN H with unity and satisfies IFP, then H has the strong IFP iff H is a PSN.

In this paper the terms, regular near-rings,r-regular near-rings, symmetric near-ring, weakly regular near-ring, completely prime ideal, 1-prime ideal, 1-semiprime ideals are introduced. We investigated some basic properties for r-regular near-rings. We use completely prime ideal, maximal ideal, 1-prime ideal, 1-semiprime ideals to characterize r-regular near-rings. Finally, we proved that the following conditions concerning for r-regular near-ring with identity and has IFP are equivalent (1) N is regular near-ring. (2) A = for every N-subgroup A of N. (3) N is left bipotent. (4) N is strongly regular near-ring. And also it is proved that the following conditions concerning for a near-ring N with identity are equivalent (1) N is r-regular and has IFP and (2) N is reduced and every completely prime ideal is maximal.

Tamkang Journal of Mathematics

We introduce the notion of left prime weakly regular, left prime weakly π-regular and left prime pseudo π-regular near-rings. We also introduce the concept of strong left prime weakly regular near-rings. We obtain conditions for a near-ring N to be left prime pseudo π-regular. We also obtain conditions for a near-ring N to be strong left prime pseudo π-regular. Finally we answer an open question given by G. F. Birkenmeier and N. J. Groenewald [Math. Pannonica 10, No. 2, 257-269 (1999; Zbl 0963.16047)].

Communications of the Korean Mathematical Society, 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 Pstrongly 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 nearring N , the following holds: (i) N a + P is an ideal of N for any a ∈ N. (ii) Every P-prime ideal of N containing P is maximal. (iii) Every ideal I of N fulfills I + P = I 2 + P .

International Research Journal of Pure Algebra, 2014

In ([5]) we defined a right near -ring N to be 𝛽𝛽 1 (𝛽𝛽 2 ) if xNy = Nxy(xNy = xyN) for all x, y in N. Following these we make an attempt in this paper to study the properties of those near -rings which satisfy the conditions xNy = yxN and xNy = Nyx.

Journal of Mathematical and Fundamental Sciences

In this paper, assuming that is a near-ring and is an ideal of , the-center of , the-center of an element in , the-identities of are defined. Their properties and relations are investigated. It is shown that the set of allidentities in is a multiplicative subsemigroup of. Also,-right and-left permutable and-medial near-rings are defined and some properties and connections are given.-regular and-strongly regular near-rings are studied.-completely prime ideals are introduced and some characterizations ofcompletely prime near-rings are provided. Also, some properties ofidempotents,-centers,-identities incompletely prime near-rings are investigated. The results that were obtained in this study are illustrated with many examples.

Journal of Pure and Applied Algebra, 2007

A ring R is called left morphic if R/Ra ∼ = l(a) for every a ∈ R. A left and right morphic ring is called a morphic ring. If M n (R) is morphic for all n ≥ 1 then R is called a strongly morphic ring. A well-known result of Erlich says that a ring R is unit regular iff it is both (von Neumann) regular and left morphic. A new connection between morphic rings and unit regular rings is proved here: a ring R is unit regular iff R[x]/(x n ) is strongly morphic for all n ≥ 1 iff R[x]/(x 2 ) is morphic. Various new families of left morphic or strongly morphic rings are constructed as extensions of unit regular rings and of principal ideal domains. This places some known examples in a broader context and answers some existing questions.

We show some characteristics of quasiideals of a P-regular near-ring and in particular, we consider the representations of elements of quasiideals of a P-regular near-ring related with the ideal P. As a result, it is proved that every element of a quasiideal Q of a P-regular near-ring can be represented as the sum of two elements of P and Q. Equivalent conditions are obtained for a near-ring to be P-regular near-ring.

International Journal of Pure and Apllied Mathematics, 2013

In this paper, we introduce the notion of strong IFP and weak IFP near-rings. Weak IFP near-ring is a generalization of IFP near-ring. We study the basic properties of right weak IFP near-rings. We show that if N is a 2-primal near-ring and if N is strong IFP, then N is left weakly regular if and only if every prime ideal of N is maximal.

Zenodo (CERN European Organization for Nuclear Research), 2021

Highlights • This paper focuses on ideal theory of near-rings. • Classical algebraic substructures of near-rings are introduced in this study. • Highly useful results are obtained about the characterizations of near-rings.