The Auslander-Reiten conjecture for certain non-Gorenstein Cohen-Macaulay rings

The Auslander-Reiten conjecture is a notorious open problem about the vanishing of Ext modules. In Cohen-Macaulay complete local ring R with parameter ideal Q, holds for if and only it residue R/Q. former part this paper, we study R/Qℓ in connection that R, prove equivalence them case where Gorenstein ℓ≤dim⁡R. latter part, generalize result minimal multiplicity by J. Sally. Due to these two our results, see there exists an Ulrich whose intersection. We also explore determinantal rings.