Local rings of minimal length

This paper deals with local rings RR possessing an mm-canonical ideal ωω, R⊆ωR⊆ω. In particular those rings such that the length lR(ω/R)lR(ω/R) is as short as possible are studied. The same notion for one-dimensional local Cohen–Macaulay rings was introduced and studied with the name of Almost Gorenstein. Some necessary conditions, that become also sufficient under additional hypotheses, are given and examples are provided also in the non-Noetherian case. The case when the maximal ideal of RR is stable is also studied.