In this paper, we make the notion of approximating an Artinian local ring by a Gorenstein Artin local ring precise using the concept of Gorenstein colength. We also answer the question as to when the Gorenstein colength is at most two.
Given an Artinian local ring $R$, we define 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 $R = T/I$ satisfies the inequality $g(R) leq lambda(R/soc(R))$ in t
he following two cases: (a) $T$ is a power series ring over a field of characteristic zero and $I$ 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 $I$ 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 $n$th 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. Keywords: Gorenstein colength; Gorenstein linkage.
