Distributively generated trivial near-rings (Q1271930)

Let \(G\) be an additively written group, \(S\) a subset of \(G\) containing \(0\) and \(T(G,S)\) the trivial near-ring on \(G\) with multiplication \(ab=0\) if \(a\in S\) and \(ab=b\) if \(a\notin S\) for \(a,b\in G\). \(T(G,S)\) is a distributively generated near-ring if and only if \(S\) is a subgroup of \(G\) of index \(2\).