Regular Elements and Von-Neumann Inverses of a Class of Zero Symmetric Local Near-Rings Admitting Frobenius Derivations

Abstract

Let N be a zero-symmetric local near-ring. An element \({x}\) \(\in\) N is either regular, zero or a zero divisor. In this paper, we construct a class of zero symmetric local near-ring of characteristic pk; k \(\ge\) 3 admitting an identity frobenius derivation, characterize the structures and orders of the set R(N), the regular compartment with an aim of advancing the classication problem of algebraic structures. The number theoretic notions relating the number of regular elements to Euler's phi-function and the arithmetic functions of Galois near-rings are adopted. Using the Fundamental Theorem of nitely generated Abelian groups, the structures of R(N) are proved to be isomorphic to cyclic groups of various orders. The study also extends to the automorphism groups Aut(R(N)) of the regular elements.