On the ideal case of a conjecture of Auslander and Reiten

A celebrated conjecture of Auslander and Reiten claims that a finitely generated module M that has no extensions with M⊕Λ over an Artin algebra Λ must be projective. This conjecture is widely open in general, even for modules over commutative Noetherian local rings. Over such rings, we prove that a large class of ideals satisfy the extension condition proposed in the aforementioned conjecture of Auslander and Reiten. Along the way we obtain a new characterization of regularity in terms of the injective dimensions of certain ideals.