A sufficient condition for F-purity

It is well known that nice conditions on the canonical module of a local ring have a strong impact in the study of strong F-regularity and F  -purity. In this note, we prove that if (R,m)(R,m) is an equidimensional and S2S2 local ring that admits a canonical ideal I≅ωRI≅ωR such that R/IR/I is F-pure, then R is F-pure. This greatly generalizes one of the main theorems in [2]. We also provide examples to show that not all Cohen–Macaulay F-pure local rings satisfy the above property.