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The present volume is the Proceedings of the 18th International Conference on Nearrings and Nearfields held at the Helmut-Schmidt-Universitat, Universitat der Bundeswehr Hamburg, from July 27 – August 3, 2003. It contains the written versions of the lectures by the five invited speakers. These concern recent developments of planar nearrings, nearrings of mappings, group nearrings and loop-nearrings. One of them is a long and very substantial research paper "The Z-Constrained Conjecture". They are followed by 13 contributions reflecting the diversity of the subject of nearrings and related structures. Besides the purely algebraic structure theory these papers show many connections of nearring theory with group theory, combinatorics, geometries, and topology. They all contain original research.

This paper consists essentially of a table showing for all 862 non-abelian groups G of order |G| ≤ 100 the order of the inner automorphism nearring (I(G), +, ∘) and the order of the commutator subgroup (I(G)’, +) of (I(G), +). The table was computed by using a program in GAP (Groups, Algorithms and Programming). The extensive calculations were done on a SUN-SPARCstation 20.

After a historical introduction this report announces in detail seventeen theorems about endomorphism nearrings on finite groups for various groups and classes of groups, mainly without proofs or only sketching some proofs, but indicating instead which theorems with their full proofs we plan to publish together in one paper and what headings these papers will have.

Publisher Summary This chapter discusses the bilinearly generated near-algebras. A bilinearly generated near-algebra is logically between a near-algebra over a near-ring and algebra over a ring. The chapter presents some theorems concerning these bilinearly generated near-algebras. A distributively generated near-ring can be derived as a special case of a bilinearly generated near-algebra. It is shown that the endomorphisms of a near-module generate a bilinearly generated near-algebra. The chapter defines the notions of a near-ring, a near-module over a near-ring, and a near-algebra over a near-ring and presents some examples of each. The relations of these algebraic structures to each other and to the well-known structures of a ring, a module over a ring, and algebra over a ring are explained.