case that $S$ is Gorenstein, the integer $c_S(R)$ is used to estimate homological properties of $R$, for example, see [@Kustin-Vraciu Theorem 7.5]. Ananthnarayan [@Ananthnarayan] introduced the [*Gorenstein colength*]{} $g(R)$ of an artinian local ring $(R,\m,k)$ to be the following integer $$g(R)\coloneqq\min\{c_S(R) \mid S \text{ is a Gorenstein artinian local ring mapping onto } R\}.$$

The number $g(R)$ measures how close $R$ is to a Gorenstein ring. Clearly, $g(R)$ is zero if and only if $R$ is Gorenstein. One can see that $g(R)=1$ if and only if $R$ is non-Gorenstein and $R\cong S/\mathsf{soc}(S)$ for an artinian Gorenstein ring $S$. These rings are called *Teter rings*. On Teter rings, the following characterization is known, which is an improvement of Teter’s result [@Teter]. This was proved by Huneke-Vraciu [@Huneke-Vraciu] under the assumption that $1/2\in R$ and $\mathsf{soc}(R)\subseteq\m^2$, and later Ananthnarayan-Avramov-Moore [@A-A-M] removed the assumption $\mathsf{soc}(R)\subseteq\m^2$. See also the result of Elias-Takatsuji .

\[Huneke-Vraciu, Ananthnarayan-Avramov-Moore, Elias-Takatuji\] \[thHVAAM\] Let $(R,\m,k)$ be an artinian local ring such that either $R$ contains $1/2$ or $R$ is equicharacteristic with $\mathsf{soc}(R)\subseteq \m^2$. Then the following are equivalent.

-   $g(R)\leq 1$.

-   Either $R$ is Gorenstein or $\m\cong \m^\dag$.

-   Either $R$ is Gorenstein or there exists a surjective homomorphism 