Structure Theorems for the Socle-Ideal of a Near-Ring

Abstract

For d.g. near-rings R satisfiying the descending chain condition for left ideals, Laxton and Machin (1972) used a faithful R-group to construct an ideal, which they called the critical ideal. The critical ideal turns out to be the socle-ideal of the d.g. near-ring R and its construction is based on the classification of the isomorphism classes of R-groups of type-0. We consider more general zero symmetric near-rings and although our method of classification is quite different from the Laxton–Machin one, we end up with the same isomorphism classes of R-groups of type-0. The socle-ideal is realized as a direct sum of copies of R-groups of type-0 from one of the classes. We then apply our theory to factor near-rings arising from nil-rigid series and to matrix near-rings.