Quotient nearrings of semilinear nearrings (Q1806245)

Let \(\mathbb{R}\) denote the field of reals. Given a continuous map \(\lambda\colon\mathbb{R}^n\to\mathbb{R}\) put \(v*w:=\lambda(v)w\). Then \(N_\lambda(\mathbb{R}^n)=(\mathbb{R}^n,+,*)\) is a topological nearring iff \(\lambda(av)=a\lambda(v)\) for all \(a\) in the range of \(\lambda\), and for all \(v\in\mathbb{R}^n\). Every ideal \(J\) of \(N_\lambda(\mathbb{R}^n)\) is also a vector subspace of \(\mathbb{R}^n\). The author proves that for all ideals \(J_1,J_2\) of \(N_\lambda(\mathbb{R}^n)\) the quotient nearrings \(N_\lambda(\mathbb{R}^n)/J_1\) and \(N_\lambda(\mathbb{R}^n)/J_2\) are isomorphic iff \(\dim J_1=\dim J_2\). Some examples are also given.