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Acta Mathematicae Academia Seientiarum Hungarieae Tomus 26 (1--2), (1975), 87--90. By D. A. LAWVER (Tucson) In [1], ANSHEL and CLAY discuss planar algebraic systems with applications to geometry. The purpose of this paper is to indicate that all non-trivial near-rings on the cocyclic groups Z(p ~) (with at least three inequivalent left multipliers) are planar and, by using the techniques of G. FERRERO [4], indicate that integral planar near- rings are definable on cocyclic groups. 1. Preliminaries A near-ring is a triple (N, +, .) where (N, +) is a group (N, .) a semi-group, and a.(b+c)=(a.b)+(a.c) for all a,b, cCN. It is well known that defining a near-ring on a group (N, +) is equivalent to fixing a function f: N~End (N) where a.b=(f(a))(b) and f(a.b)=f(a)of(b) [3]. We will adhere to this notation throughout. Given a near-ring N defined by f: N~End (N), two elements a, bEN are said to be left equivalent multipliers, denoted a-=m b, if f (a)=f(b) or, equivalently, ax = bx for all xEN [1]. We need the following from [1]: DEFINITION 1.1. A near-ring (N, +, -) is said to be planar if i) a.x=b .x+c has a unique solution for x given
Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals
Published: May 21, 2016
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