Abstract. In this paper, we give topological properties of collection of prime ideals in 2-primal near-rings. We show that Spec(N ), the spectrum of prime ideals, is a compact space, and Max(N ), the maximal ideals of N, forms a compact T 1 -subspace. We also study the zero-divisor graph Γ I (R) with respect to the completely semiprime ideal I of N. We show that Γ P (R), where P is a prime radical of N, is a connected graph with diameter less than or equal to 3. We characterize all cycles in the graph Γ P (R). PreliminariesIn [3], Beck introduced the concept of a zero-divisor graph of a commutative ring with identity, but this work was mostly concerned with coloring of rings. In [2], Anderson and Livingston associated a graph (simple) Γ(R) to a commutative ring R with identity with vertices Z(R) * = Z(R)\{0}, the set of nonzero zero-divisor of R, and for distinct x, y ∈ Z(R) * , the vertices x, and y are adjacent if and only if xy = 0. They investigated the interplay between the ring-theoretic properties of R and the graph-theoretics properties of Γ(R).In [9], Redmond has generalized the notion of the zero-divisor graph. For a given ideal I of R, he defined an undirected graph Γ I (R) with vertices {x ∈ R\I : xy ∈ I for some y ∈ R\I}, where distinct vertices x and y are adjacent if and only if xy ∈ I.In this paper, we study the undirected graph Γ I (N ) of near-rings for any completely semiprime ideal I of N. We extend the results obtained by K. Samei [11] for reduced rings to 2-primal near-rings. Clearly, reduced rings are 2-primal near-rings.Let N be a near-ring with identity. Let J be a completely semiprime ideal of N. The zero-divisor graph of N with respect to the ideal J, denoted by Γ J (N ), is the graph whose vertices are the set {x ∈ N \J : xy ∈ J for some y ∈ N \J} with distinct vertices x and y are adjacent if and only if xy ∈ J. If J = 0, then
Human endeavours span a wide spectrum of activities which includes solving fascinating problems in the realms of engineering, arts, sciences, medical sciences, social sciences, economics and environment. To solve these problems, classical mathematics methods are insufficient. The real-world problems involve many uncertainties making them difficult to solve by classical means. The researchers world over have established new mathematical theories such as fuzzy set theory and rough set theory in order to model the uncertainties that appear in various fields mentioned above. In the recent days, soft set theory has been developed which offers a novel way of solving real world issues as the issue of setting the membership function does not arise. This comes handy in solving numerous problems and many advancements are being made now-a-days. Jun introduced hybrid structure utilizing the ideas of a fuzzy set and a soft set. It is to be noted that hybrid structures are a speculation of soft set and fuzzy set. In the present work, the notion of hybrid ideals of a near-ring is introduced. Significant work has been carried out to investigate a portion of their significant properties. These notions are characterized and their relations are established furthermore. For a hybrid left (resp., right) ideal, different left (resp., right) ideal structures of near-rings are constructed. Efforts have been undertaken to display the relations between the hybrid product and hybrid intersection. Finally, results based on homomorphic hybrid preimage of a hybrid left (resp., right) ideals are proved.
The notions of hybrid ideals and k-hybrid ideals in a ternary semiring are introduced in this paper, and a substantial amount of effort has been made to study some of their features. In terms of characteristic function, we show some properties of k-hybrid ideals and give some characterizations of hybrid intersection with respect to these k-hybrid ideals. Finally, results based on a k-hybrid ideal’s homomorphic hybrid preimage are provided. With respect to k-hybrid ideals, we give certain characterizations of hybrid intersection.
The concept of a hybrid structure in X -semimodules, where X is a semiring, is introduced in this paper. The notions of hybrid subsemimodule and hybrid right (resp., left) ideals are defined and discussed in semirings. We investigate the representations of hybrid subsemimodules and hybrid ideals using hybrid products. We also get some interesting results on t-pure hybrid ideals in X . Furthermore, we show how hybrid products and hybrid intersections are linked. Finally, the characterization theorem is proved in terms of hybrid structures for fully idempotent semirings.
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