Almost Gorenstein rings

Barucci and Fröberg (1997) [2] introduced the notion of an almost Gorenstein ring in the case of an analytically unramified local ring (R,m). In this work we provide a framework that allows us to generalize this notion to include the analytically ramified case. As a consequence, for any Cohen–Macaulay local ring (R,m) of dimension one we solve in full generality the problem of determining when the endomorphism algebra m:m is Gorenstein. We also provide characterizations for rings to be almost Gorenstein in connection with the principle of idealization. Several examples are explored as well.