A ring associated with a near-ring (Q1920012)

Let \(N\) be a right abelian near-ring with identity. The set of all right multiplication maps on \(N\) generates a ring \(R\) which is a subring of \(\text{End}(N)\). The main results are the following ones: 1. If \(N\) is a finite near-field with center \(K\) and \(\dim_KN=n\), then \(R\) is isomorphic to \(M_n(K)^K\). 2. If \(N=M_A(G)\) is a simple abelian near-ring, \(A\) is a finite group of automorphisms of \(G\), then \(R\) is isomorphic with \(M_t(S)\), where \(S\) is the subring of \(\text{End}(G)\) generated by \(A\) and \(t\) is the number of non-zero \(A\)-orbits of \(G\). 3. If \(N=M_A(G)\), where \((M_A(G),+)\) is a finite abelian group and \(A\) is a group of automorphisms of the finite group \(G\), then \(R=I+T\), where \(I\) is a nilpotent ideal of \(R\) and \(T\) is a subring of \(R\) such that \(I\cap T=\{0\}\) and \(R/I\) is isomorphic to \(T\), \(T\) being a direct sum of matrix rings.