Characterizations of a two dimensional Euclidean ring among near‐rings

Abstract

All those multiplications on the two-dimensional Euclidean group are determined such that the resulting non-associative topological nearring has (1, 0) for a left identity and has the additional property that every element of the near-ring is a right divisor of zero. This result, together with several previous results, is then used to show that any one of several common algebraic properties is sufficient to characterize one particular two-dimensional Euclidean ring within the class of all two dimensional Euclidean near-rings. Specifically, it is proved that, if N is a topological near-ring with a left identity whose additive group is the two-dimensional Euclidean group, then the following assertions are equivalent: (1) the left identity is not a right identity, (2) N contains a non-zero left annihilator, (3) every element of N is a right divisor of zero, (4) Nw≠N for all w∈N, (5) N is isomorphic to the topological ring whose additive group is the two dimensional Euclidean group and whose multiplication is given by (v1, V2)(w1W2) = (v1w1, v1w2).