𝐹-purity and rational singularity

We investigate singularities which are F F -pure (respectively F F -pure type). A ring R R of characteristic p p is F F -pure if for every R R -module M, 0 → M ⊗ R → M ⊗ 1 R M, 0 M R M \, ¹R is exact where 1 R ¹R denotes the R R -algebra structure induced on R R via the Frobenius map (if r ∈ R r R and s ∈ 1 R s \, ^1R, then r ⋅ s = r p s r s = rᵖs in 1 R