Two theorems on the vanishing of Ext

We prove two theorems on the vanishing of Ext over commutative Noetherian local rings. Our first theorem shows that, over non-regular Cohen-Macaulay local domains, there are no ideals that are both Burch and rigid. Our second theorem reformulates a conjecture of Huneke-Wiegand concerning rigid modules and highlights its relation with the celebrated Auslander-Reiten conjecture. We also discuss a byproduct of our argument which yields a generalization of a result of Araya.