1Let A be an Artin local ring. The Dilworth number of A is defined to be the supremum of p(Z) for ideals Z in A, where p is the minimal number of generators. In my paper [4], I developed some methods to compute this number when A is homogeneously graded over a field k, and especially if A is Gorenstein and ch k = 0, I showed that it is the dominating case that the Dilworth number is attained by a power of the maximal ideal. In such a case we shall say that A has the Sperner property. In the present paper we consider Artin rings (A, m), not necessarily graded, and compare the Dilworth number of A and that of Gr,(A), and it will be shown that the main result of [4] is valid without the assumption that the rings are homogeneous. Our main result is Theorem 7, where we prove that “most” Artin Gorenstein rings have the Sperner property. (Actually we prove a stronger result, which says that it has the “weak Stanley property.“) To explain the term “most,” we represent an Artin Gorenstein ring A in the