Topological nearrings whose additive groups are Euclidean

Abstract

Let (G,+) be a group with a locally compact Hausdorff topology for which the binary operation + is continuous. Those, binary operation * onG for which (G, +, *) is a topological nearring are described. In the case whereG is abelian, those binary operations * for which (G, +, *) is a topological ring are also described. Versions of these results are then obtained in the special case where the group is the topological Euclideann-group,Rn. A family of binary operations * for which (Rn, +, *)_is a topological nearring is then investigated in some detail. Most of these nearrings turn out to be planar. Their ideals are completely determined and we characterize those nearrings which are simple. The multiplicative semi-groups (Rn, *) of these nearrings are then investigated. Green's relations are completely determined and it is shown that a number of familiar properties of semigroups are equivalent for these particular semigroups. Finally, all those binary operations * for which (R, +, *) is a topological nearring are completely described. It is determined when any two of these nearrings are isomorphic and for each of these nearrings, its automorphism group, is completely determined.