Primeness in near-rings of continuous functions.
In this review, all nearrings will be right distributive. A nearring \(\mathcal N\) is `strongly prime' if for \(a\neq 0\) there exists a finite subset \(F\subseteq{\mathcal N}\) such that for all \(x\in{\mathcal N}\), \(aFx=0\) implies \(x=0\). The nearring \(\mathcal N\) is `equiprime' if for all \(a,x,y\in{\mathcal N}\), \(anx=any\) for all \(n\in{\mathcal N}\) implies either \(a=0\) or \(x=y\) and it is `strongly equiprime' if for all nonzero \(a\in{\mathcal N}\) there exists a finite subset \(F\subseteq{\mathcal N}\) such that for all \(x,y\in{\mathcal N}\), \(anx=any\) for all \(n\in{\mathcal F}\) implies \(x=y\). Let \(G\) be an additive topological group. We denote by \({\mathcal N}_0(G)\) the nearring, under pointwise addition and composition, of all continuous selfmaps \(f\) of \(G\) such that \(f(0)=0\). Let \(G\) be a \(0\)-dimensional \(T_0\) topological group with more than one element. The authors prove that (1) \({\mathcal N}_0(G)\) is equiprime, (2) \({\mathcal N}_0(G)\) is strongly prime if and only if \(G\) is topologically discrete and (3) \({\mathcal N}_0(G)\) is strongly equiprime if and only if \(G\) is finite. They go on to show that if \(G\) has a base of arcwise connected open subsets, then \({\mathcal N}_0(G)\) is not strongly prime. Let \(X\) be a topological space and let \(\theta\) be a continuous map from \(G\) to \(X\). The nearring of all continuous functions \(f\) from \(X\) to \(G\) such that \(f\circ\theta(0)=0\), under pointwise addition and multiplication defined by \(fg=f\circ\theta\circ g\) is denoted by \({\mathcal N}_0(G,X,\theta)\). The authors prove that if \(X\) is a \(0\)-dimensional \(T_0\) space with more than one element and \(G\) is a \(T_0\) topological group with more than one element, then \({\mathcal N}_0(G, X,\theta)\) is equiprime if and only if \(\theta\) is injective and \(\text{cl}(\theta(G))=G\).
scientific article; zbMATH DE number 2096227