Characterizing series for faithful d.g. near-rings

Abstract:

Let (R, S) be a distributively generated near ring satisfying $(R,S) \subseteq (E(G),{\text {End}}(G))$ and $S \subseteq {\text {End}}(G)$ for some group G, endomorphism near ring $E(G)$, and subsemigroup S of the endomorphisms of G, ${\text {End}}(G)$. The radicals $J(R)$ of (R, S) are characterized in terms of series of subgroups of G. We assume S contains the inner automorphisms of G and obtain two main results on characterizing series. (1) If G satisfies both chain conditions on S subgroups then a unique minimal characterizing series exists. (2) If G is finite, then both maximal and minimal characterizing series exist, are unique, and are themselves characterized in G.