Abstract. We show that, up to isomorphism, there is exactly one topological nearring which is not zero symmetric and has an identity, whose additive group is the two-dimensional Euclidean group ${\msbm R}^2$. We determine the endomorphism semigroup and the automorphism group of this nearring. We determine its right, left and two-sided ideals. In particular, we show that it has exactly one nonzero proper two-sided ideal and that the corresponding quotient nearring is the field of real numbers. Finally, we investigate the structure of the multiplicative semigroup of this nearring.

AMS Subject Classification (1991): 16Y30

Received November 7, 1995. (Registered under 5712/2009.)