Let N denote a 3–prime zero-symmetric left near-ring, and let d be a nonzero derivation on N. Let U ≠ φ be a nonzero subset of N such that (i) U N ⊆ U or (ii) N U ⊆ U. We prove that N must be a commutative ring if one of the following holds: (a) U satisfies one of (i) and (ii), and d(U) is multiplicatively central; (b) U satisfies both of (i) and (ii), d 2 ≠ 0, and [d(U), d(U)] = {0}. Some related results are also given.

On Derivations in Near-Rings, II

A taxonomy of 2-primal rings

The analog of Posner's theorem on the composition of two derivations in prime rings is proved for 3-prime near-rings. It is shown that if d is a nonzero derivation of a 2-torsionfree 3-prime near-ring N and an element a ∊ N is such that axd = xda for all x ∊ N, then a is a central element. As a consequence it is shown that if d\ and d2 are nonzero derivations of a 2-torsionfree 3-prime near-ring N and xd1yd2 = yd2xd1 for all x, y ∊ N, then N is a commutative ring. Thus two theorems of Herstein are generalized

Posner and herstein theorems for derivations of 3-prime near-rings

Nearrings, Nearfields and K-Loops

Nice error bases are a fundamental primitive of quantum information processing. For example, they govern the discretization of errors in quantum error-correcting codes. It is show that the generalized Pauli basis, the most widely used example of nice error bases, has some remarkable structural properties. However, the generalized Pauli basis is limited to dimensions that are a power of a prime, since it is constructed with the help of a finite field. A wider class of nice error bases is introduced that shares many features of the generalized Pauli basis, yet allows one to remove the restriction to prime power dimensions. The nice error bases are indexed by nearrings. Nearrings that support the construction of nice error bases are called nice. It is shown that all finite nearfields are nice. It is shown that a finite ring is nice if and only if it finite Frobenius ring. Several fundamental properties of nice nearrings are established.

http://ieeexplore.ieee.org/iel5/6268627/6281806/06283569.pdf

Nice nearrings

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.

Günter E. Pilz

Papers

Near-rings and Near-fields