Given an Artinian local ring R, we define (in [1]) its Gorenstein colength g(R) to measure how closely we can approximate R by a Gorenstein Artin local ring. In this paper, we show that R = T/b satisfies the inequality g(R) ≤ λ(R/soc(R)) in the following two cases: (a) T is a power series ring over a field of characteristic zero and b an ideal that is the power of a system of parameters or (b) T is a 2- dimensional regular local ring with infinite residue field and b is primary to the maximal ideal of T. In the first case, we compute g(R) by constructing a Gorenstein Artin local ring mapping onto R. We further use this construction to show that an ideal that is the nth power of a system of parameters is directly linked to the (n — 1)st power via Gorenstein ideals. A similar method shows that such ideals are also directly linked to themselves via Gorenstein ideals.
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