Linear right ideal nearrings (Q5957248)

Let \(N:=(\mathbb{R}^n,+)\), the additive, topological group of \(n\) dimensions over the reals. The author studies multiplications on \(N\) such that \(N\) becomes a topological (right) nearring. In the present paper, he requires the following additional property: \(N\) contains \(n\) right ideals \(J_1,\dots,J_n\) which are 1-dimensional subspaces of \(N\) such that every \(w\in N\) has a representation \(w=w_1+\cdots+w_n\), \(w_i\in J_i\), and \(vw=vw^n\) for all \(v\in N\). The author gives explicit formulae for the possible multiplications on \(N\) with these properties. These results are specialized to the case \(n=2\).