Topological nearrings whose additive groups are Euclidean
The author is concerned here with additive groups with a locally compact Hausdorff topology for which the group operation is continuous. The conditions under which a multiplication can be defined on the group which makes it into a nearring or a ring are determined. After this general introduction, the special case where the group is the topological Euclidean \(n\)-group \(R^n\) is considered in detail. A family of nearrings defined on these groups are analysed in some detail. Most of the nearrings turn out to be planar. Their ideals are characterized. The multiplicative semigroups of the nearrings are considered in detail, and they are shown to be of a particularly well-behaved kind. The lattice of congruences is also determined. Finally the special case of \(R\) itself is considered. The detailed structure of the corresponding nearrings is obtained including the automorphism group.
scientific article; zbMATH DE number 757759