Topological N-groups on the Reals

K. D. Magill, Jr., Buffalo, USA

Keywords: Topological nearrings, topological N-groups, ideals AMS Subject classification: 16Y30, 16W80, 54H13

Abstract: Let N be a topological nearring and let G be a topological group. Suppose there exists a continuous map from N × G which sends (n, g) to ng ∈ G such that (n_{1} + n_{2})g = n_{1}g + n_{2}g and (n_{1}n_{2})g = n_{1}(n_{2}g) for all n_{1}, n_{2} ∈ N and g ∈ G. Then G, together with this multiplication on the left by elements of N, is referred to as a topological N-group or topological nearmodule. In this paper, N will be a topological nearring whose additive group is a Euclidean N-group and G will be the additive group R of real numbers. Within this class of nearrings, we completely determine those nearrings N for which there exist nontrivial multiplications such that R is a topological N-group and we determine precisely when two of these nearrings are isomorphic.