Near Rings, Fuzzy Ideals, and Graph Theory

Some New Results on Near Fields

“…Remark 2.3 Every ring and near field are 𝐶-rings. Definition 2.4 (Regular Near Ring) [21] If for each element x of a near ring 𝑁, there is an element 𝑦 in 𝑁 such that 𝑥 = 𝑥𝑦𝑥, then 𝑁 is called a regular near ring. The regular rings are generalisations of division rings and notice that not every regular ring is a division ring.…”

Adding Rows in Main e-Abacus Diagrams

“…The set {𝛽 1 , 𝛽 2 , ⋯ 𝛽 b } is said to be the set of 𝛽 -number for 𝜏, see [3]. Let e be a positive integer number greater than or equal to 2, we can represent numbers by a diagram called e-abacus diagram, see [4], as shown in Table 1: Where every 𝛽numbers will be represented by (▪) which takes its location in e-abacus diagram and in case of nonexistence of 𝛽 -numbers, then it will be represented by (-). From the above example (6, 6, 3, 3, 2, 1):…”

Graph of an Intuitionistic Fuzzy Ideal of M⟙ Groups in Near Rings

“…As near ring has their application in automation, topology and ideal in order theory and graph theory, a graph is extended to fuzzy graph and discussed their applications in decision process also, by Rosenfeld et al [11] in 1975. Sathyanarayana and Prasad et al [12] expanded near rings with ideals of fuzzy and theoretical idea of graph theory by explaining theorems and representing ideals into graphs.…”