  A celebrated conjecture of Auslander and Reiten claims that a finitely
generated module $M$ that has no extensions with $M\oplus \Lambda$ over an
Artin algebra $\Lambda$ 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.
