Planar Division Neo-rings

Introduction. The notion of a division ring can be generalized to give a system whose addition is not necessarily associative, but which retains the property of coordinatizing an affine plane. Such a system will be called a planar division neo-ring (PDNR); examples of (infinite) PDNRs which are not division rings are known. If (R, +, •) is a finite power-associative PDNR, then (R, +) is shown to be commutative and to possess the inverse property. The center of an arbitrary PDNR, and the nucleus of a finite PDNR, are shown to be PDNRs. By means of these and similar properties it is demonstrated that all associative PDNRs of order =250 are actually abelian. The main result is the following: if (R, +, ■) is a finite associative and commutative PDNR of order n, and if p is any prime dividing n, then the mapping x—>xp is an automorphism of (R, +, ■). Chiefly by means of this result, all associative and commutative PDNRs of order ^250 are shown to have prime-power order. Chapter I contains results about the planar ternary rings developed by Marshall Hall [7], with a sketch of their connection with the complete sets of orthogonal latin squares associated with affine planes. Chapter II is devoted to strictly algebraic theory of PDNRs, mostly for the finite case. Chapter III contains the main theorem about automorphisms mentioned above, and examples of its application. In the Appendix will be found examples of infinite PDNRs which are not division rings. These results are from the author's doctoral dissertation at the University of Wisconsin; the author wishes to take this opportunity to express his gratitude to Professor R. H. Bruck for invaluable assistance in carrying out this research.