Matrix near-rings contained in 2-primitive near-rings with minimal subgroups (Q1188157)

Matrix near-rings as introduced by the author and the reviewer [Arch. Math. 47, 312-319 (1986; Zbl 0611.16025)] have been studied mainly with a view to the transfer of properties between the near-ring and the matrix near-ring. Here the author shows how they occur naturally. The main result is as follows. Let \(G\) be a group, \(A=D\cup\{0\}\), where \(D\) is a fixed point free group of automorphisms of \(G\). Then \((G,A)\) is a finite dimensional near vector space in the sense of \textit{J. André} [Math. Z. 136, 295-313 (1974; Zbl 0271.15001)] if and only if \(M_ A(G)\) is a near-ring with a complete set of distributive idempotents, i.e. a finite set of distributive idempotents \(\{e_ 1,\dots,e_ n\}\) such that \(e_ 1+\dots+e_ n=1\), \(e_ i e_ j=0\) if \(i\neq j\), and \(\text{rank }e_ i=1\) for all \(i\). If \((G,A)\) is \(n\)-dimensional, then \(M_ A(G)\) contains as a dense subnear-ring a near-ring isomorphic to a generalized matrix near-ring. There are several other interesting results in this paper.