Suppose R R is a F F -finite and F F -pure Q \mathbb {Q} -Gorenstein local ring of prime characteristic p > 0 p>0 . We show that an ideal I ⊆ R I\subseteq R is uniformly compatible ideal (with all p − e p^{-e} -linear maps) if and only if exists a module finite ring map R → S R\to S such that the ideal I I is the sum of images of all R R -linear maps S → R S\to R . In other words, the set of uniformly compatible ideals is exactly the set of trace ideals of finite ring maps.