Generalized Derivations on Prime Near Rings
Let be a near ring. An additive mapping is said to be a
right generalized (resp., left generalized) derivation with associated derivation on if (resp., ) for all . A mapping
 is said to be a generalized derivation with associated derivation on if
 is both a right generalized and a left generalized derivation with associated derivation
 on . The purpose of the present paper is to prove some theorems in the setting of
a semigroup ideal of a 3-prime near ring admitting a generalized derivation, thereby
extending some known results on derivations.
On Prime Near-Rings with Generalized Derivation
Let 𝑁 be a 3-prime 2-torsion-free zero-symmetric left near-ring with multiplicative center 𝑍. We prove that if 𝑁 admits a nonzero generalized derivation 𝑓 such that 𝑓(𝑁)⊆𝑍, then 𝑁 is a commutative ring. We also discuss some related properties.
Near-rings and Near-fields
Near-rings are generalized rings: commutativity of addition is not assumed, and 3=more important — only one distributive law is required. The most famous example is the collection of all maps from an additive group into itself w.r.t. function addition and composition. Compared with a standard class of rings, endomorphism rings of abelian groups, one sees that rings describe “linear” maps on groups, while near-rings handle the general nonlinear case. The theory of near-rings is now a sophisticated theory which has found numerous applications in various areas. In this article, we concentrate on a deep structure theorem for near-rings, the density theorem for primitive near-rings. This result is applied to interpolation theory, group theory and polynomials. Connections between other parts of near-rings (especially near-fields) and geometry come up at several places. Efficient block designs and codes can be constructed from finite near-rings. Finally we mention connections to ring theory, computer science, graph theory and other parts of the “outside world”. Only some easier proofs are given. For the others, references will be given; if these results are available in book form, we cite this book and not the original paper. Up to now, three books on near-rings and one on near-fields have appeared (see [1-4]in the list of references). , , o denote the sets if natural numbers, integers and reals, respectively, and Zn denotes the set of residue classes modulo n.