This paper complements another paper by the same author [ibid. 93, 105- 110 (1982; Zbl 0506.16029)]. Zero-symmetric right near-rings are considered. The socle-ideal of such a near-ring N is defined as the maximal ideal (if it exists) which is the direct sum of left ideals of the form \(Ne_ i\), all of type 0, \(e^ 2_ i=e_ i\) and \(e_ ie_ j=0\) if \(i>j\) for some ordering on the indices. \((Ne_ i\) is of type 0 if it is a minimal left ideal.) Denote it by Soi(N). It is either \(\{\) 0\(\}\) or it exists and has trivial intersection with the intersection of all maximal modular left ideals, hence annihilates \(J_ 0(N)\) the smallest of the Jacobson type radicals commonly studied in near-rings. With the addition of suitable descending chain conditions, a number of interesting structural results are proved. The connection of Soi(N) with the crux of N, defined by S. D. Scott [Proc. Lond. Math. Soc., III. Ser. 25, 441-464 (1972; Zbl 0253.16026)] is investigated, as is its links with the \(J_ s\) radical defined in the author’s paper referred to above.