ideal of an $R$-module and nearly Gorensteiness of local rings (see [@HHS]).

\[deftr\] Let $R$ be a commutative ring. For an $R$-module $M$, the [*trace ideal*]{} $M^*(M)$ of $M$ in $R$ is defined to be the ideal $\sum_{f\in \Hom_R(M,R)} \image f\subseteq R$.

A Cohen-Macaulay local ring $(R,\m)$ with a canonical module $\omega$ is called [*nearly Gorenstein*]{} if $\omega^*(\omega)\supseteq \m$.

\[lemng\] Let $(R,\m)$ be a Cohen-Macaulay local ring with a canonical module. The following are equivalent.

-   $R$ is nearly Gorenstein.

-   there is a surjective homomorphism $\omega^{\oplus n} \to \m$ for some $n$.

Moreover, if $\dim R\leq 1$, then we can add the following conditions.

-   there is a short exact sequence $0 \to \m^\dag \to R^{\oplus n} \to M \to 0$ for some $n$ and maximal Cohen-Macaulay module $M$.

-   there is a short exact sequence $0 \to \m^\dag \to \m^{\oplus n} \to M \to 0$ for some $n$ and maximal Cohen-Macaulay module $M$.

\(1) $\iff$ (2): By the definition of trace ideals, there is a surjection $\omega^{\oplus n}\to \omega^*(\omega)$ for some $n$. So the equivalence immediately follows.

Now assume $\dim R\le 1$. Then the maximal ideal $\m$ is maximal Cohen-Macaulay as an $R$-module. So the condition (2) is 