Almost Gorenstein rings

The notion of almost Gorenstein ring given by Barucci and Fr{"o}berg \cite{BF} in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified. As a sequel, the problem of when the endomorphism algebra $\text{m}:\text{m}$ of $\text{m}$ is a Gorenstein ring is solved in full generality, where $\text{m}$ denotes the maximal ideal in a given Cohen-Macaulay local ring of dimension one. Characterizations of almost Gorenstein rings are given in connection with the principle of idealization. Examples are explored.