$\omega \to \m$.

Moreover, Ananthnarayan [@Ananthnarayan] extended this theorem to the case $g(R)\leq 2$ as follows.

\[Ananthnarayan\] \[Anan\] Let $(R,\m)$ be an artinian local ring. Write $R\cong T/I$ where $(T,\m_T)$ is a regular local ring and $I$ is an ideal of $T$. Suppose $I\subseteq \m_T^6$ and $1/2\in R$. Then the following are equivalent.

-   $g(R)\leq 2$.

-   There exists a self-dual ideal $\mathfrak{a}\subseteq R$ such that $l(R/\mathfrak{a})\leq 2$.

In this paper, we try to extend the notion of Gorenstein colengths and the above results to the case that $R$ is a one-dimensional Cohen-Macaulay local ring.

For a local ring $(R,\m)$, we denote by $Q(R)$ the total quotient ring of $R$. An extension $S\subseteq R$ of local rings is called [*birational*]{} if $R\subseteq Q(S)$. In this case, $R$ and $S$ have same total quotient ring.

Let $(S,\mathfrak{n})\subseteq (R,\m)$ be an extension of local rings. Suppose $\mathfrak{n}=\m\cap R$. Then $S\subseteq R$ is called [*residually rational*]{} if there is an isomorphism $S/\mathfrak{n}\cong R/\m$ induced by the natural inclusion $S\to R$. For example, if $S\subseteq R$ is module-finite and $S/\mathfrak{n}$ is algebraically closed, then it automatically follows that $S\subseteq R$ is residually rational. We introduce an invariant $bg(R)$ for local rings $R$ as follows, 