Generalized \((m,n)\) bi-ideals of a near-ring (Q1860610)

Let \(N\) be a zero-symmetric \(S_k\) (right) near-ring (\(k\geq 2\)), i.e. \(x\in Nx^k\) for all \(x\in N\). The main result of the paper is to characterize when \(N\) is a generalized near-field using various near-ring properties like sub-commutativity, \(P_k(r,m)\) near-rings, \(P_k'(r,m)\) near-rings, bi-potent, stability, regularity, near-rings with property \((\alpha)\), etc. One has to be aware that some of the terminology used in this paper may have appeared under different names by other authors.