An expanded group (sometimes called an \(\Omega\)-group) is a triple \(\langle V,+,F\rangle\), where \((V,+)\) is a group (not necessarily abelian), together with an additional set \(F\) (which is allowed to be empty) of operations on \(V\). A (left) near-ring is a triple \((N,+,\cdot)\) that satisfies the axioms for a ring, except perhaps the commutativity of addition, and the right distributive law. If \((V,+)\) is a group and \(N\) is a left near-ring, then \(V\) is called a (right) near-ring module over \(N\) if there is a mapping \(\theta : V\times N \rightarrow V\), where we write \(vn\) for \((v,n)\theta\), such that \(v(n_1+n_2) = vn_1 + vn_2\) and \(v(n_1n_2) = (vn_1)n_2\) for all \(v\in V\) and all \(n_1,n_2\in N\). Any \(N\)-module \(V\) is an expanded group \(\langle V,+,N\rangle\) where the actions of the elements of \(N\) on \(V\) are viewed as unary operations on \(V\).
In this paper, the authors focus mainly on the three cases where the near-ring \(N\) is one of \(M_0(V)\) (the near-ring of \(0\)-preserving functions of the group \((V,+)\)), \(P_0(V)\) (the near-ring of \(0\)-preserving polynomial functions of \(V\)), and \(C_0(V)\) (the near-ring of \(0\)-preserving and congruence preserving functions of \(V\)).
Several results are proven with respect to the \(C_0(V)\)-solvability and the \(C_0(V)\)-nilpotency of \(V\). An \(N\)-module \(V\) is said to be \(N\)-solvable (\(N\)-nilpotent) if there exists a series \(0 = V_0 < V_1 < \cdots < V_n=V\) of \(N\)-ideals of \(V\) such that \(V_i \leq C_V(V_i/V_{i-1})\) (\(V_i = C_V(V_i/V_{i-1})\)) for each \(i=1,2,\ldots,n\). The notation \(C_V(X)\), for \(X\subset V\), denotes the union of all \(N\)-subgroups \(H\) of \(V\) such that \((x+h)\alpha = x\alpha + h\alpha\) for all \(x\in X,\ h\in H\), \(\alpha \in N\), and called the centralizer of \(X\) with respect to \(N\).
The final sections are concerned with type preservation and solvability/nilpotency. Here, the discussions revolve around so-called tame triples. A tame triple \((N_1,N_2,V)\) consists of near-rings \(N_1 \leq N_2\) and a group \(V\) that is a faithful tame module for each of \(N_1\) and \(N_2\) with the additional property that the \(N_1\)- and \(N_2\)-ideals of \(V\) coincide. (An \(N\)-module \(V\) is tame if each \(N\)-subgroup of \(V\) is an \(N\)-ideal of \(V\).) The concept of type preservation with respect to tame triples was introduced in [G. L. Peterson and S. D. Scott, Quaest. Math. 44, No. 7, 923–944 (2021; Zbl 1493.16057)]. It is proved that if \((N_1,N_2,V)\) is a tame triple for which \(V\) is \(N_1\)-solvable and \(N_1/J_2(N_1)\) has the DCC on right ideals, with \(J_2(N)\) (the \(J_2\)-radical of \(N\)) nilpotent, then, if \((N_1,N_2,V)\) is type preserving, then \(V\) is \(N_2\)-solvable.
Another result concerns \(2\)-tame \(N\)-modules. An \(N\)-module \(V\) is \(2\)-tame if for each \(v_1,v_2,w \in V\), and for each \(\alpha \in N\), there is a \(\beta \in N\) such that \((w+v_i)\alpha - w\alpha = v_i\beta\) for each \(i=1,2\). It is shown that if \(V\) is a faithful \(2\)-tame \(N\)-module where \(N/J_2(N)\) has the DCC on right ideals, with \(J_2(N)\) nilpotent, then, if \(V\) is \(C_0(V)\)-nilpotent, then the triple \((N,C_0(V),V)\) is type preserving. The paper is concluded by an example to show that the converse of this result is not true.