Two dimensional nilpotent Euclidean nearrings (Q699911)

A nearring \(N\) is called nilpotent if there exists a natural number \(k\) such that \(N^k=0\). The least such number is called the (nilpotent) rank of \(N\). The author determines all nilpotent topological (right) nearrings of rank~\(3\) with additive group \((\mathbb{R}^2,+)\), the additive, topological group of \(2\) dimensions over the reals. Every such nearring is isomorphic to one with multiplication given by \(vw=(v_2f(w),0)\), where \(f\colon\mathbb{R}^2\to\mathbb{R}\) is a nonzero continuous map with \(f(x,0)=0\). Isomorphisms among these are determined as well. Examples show that there are infinitely many isomorphism classes of such nearrings. Ideals, right and left ideals are determined. Earlier results of the author [Monatsh. Math. 119, No. 4, 281-301 (1995; Zbl 0830.16032)] show that there does not exist a rank~3 topological nearring with additive group \((\mathbb{R},+)\). In contrast to that, there exist rank~3 nilpotent topological nearrings with additive group \((\mathbb{R}^n,+)\) for all \(n>1\).