When is $\mathfrak{m}:\mathfrak{m}$ an almost Gorenstein ring?

Given a one-dimensional Cohen-Macaulay local ring $(R,\mathfrak{m},k)$, we prove that it is almost Gorenstein if and only if $\mathfrak{m}$ is a canonical module of the ring $\mathfrak{m}:\mathfrak{m}$. Then, we generalize this result by introducing the notions of almost canonical ideal and gAGL ring and by proving that $R$ is gAGL if and only if $\mathfrak{m}$ is an almost canonical ideal of $\mathfrak{m}:\mathfrak{m}$. We use this fact to characterize when the ring $\mathfrak{m}:\mathfrak{m}$ is almost Gorenstein, provided that $R$ has minimal multiplicity. This is a generalization of a result proved by Chau, Goto, Kumashiro, and Matsuoka in the case in which $\mathfrak{m}:\mathfrak{m}$ is local and its residue field is isomorphic to $k$.

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