ON REGULAR ELEMENTS AND VON-NEUMANN INVERSES OF ZERO-SYMMETRIC LOCAL NEAR-RINGS WITH JORDAN IDEALS ADMITTING FROBENIUS DERIVATIONS

Abstract

The study of near-rings with identity is very vital in generalizing characterization of commutative rings with identity. Much of the recent works on the classification of finite rings with identity have considered a characterization paradigm using the unit groups, the zero divisor graphs, adjacency and incidence matrices among others. This has left the non-linear aspects fairly untouched. In particular, regular elements and Von-Neumann inverses of near rings admitting derivations hardly exist in literature. Thus, the study determined the structures of classes of zero symmetric local near rings N with n−nilpotent radical of Jordan ideals, J(N); n = 2, n ≥ 3 with char N as p, p2 and pk; k ≥ 3 convoluted with Frobenius derivations, the commutation over N constructed and finally characterized N,R(N), Γ(N) and the inverses of N. To achieve these, the research used idealization of R0-modules with respect to Galois rings and Raghavendran’s characterization method to construct the classes of near-rings under investigation, the theorems of Asma and Inzamam to determine the commutation over N via J(N) and the Frobenius derivations, the fundamental theorem of finitely generated abelian groups to determine the structures of R(N) and their inverses and SONATA. The results of this study showed two constructions of classes of zero symmetric local near rings with a Jordan ideal containing a 2-nilpotent radical which admit a commuting Frobenius derivation, determined some graph morphisms which form symmetric groups, the regular elements obtained have structures isomorphic to cyclic groups. The Von-Neumann inverses of the N formulated agreed with the number theoretic standards of the Von-Neumann inverses of idealized local rings while the arithmetic function, V (| R(N) |) followed the asymptotic properties of V (n), τ (n), ω(n), σ(n) and K(n). Furthermore, the results determined the automorphisms of R(N both in terms of structures and orders.