Any ring R of characteristic p > 0 is equipped with many natural endomorphisms: for e > 0, the e-th Frobenius map is given by (F^e)(r) = r^(p^e). In this talk, we'll study Hom_R(F^e_R, R): the R-module of splittings of this map. When R is a regular ring, or more generally a Gorenstein ring, this R-module can be canonically identified with F^e_R itself. When S = R/I is a quotient of a regular ring, the splittings of F^e: R/I -> R/I can be understood in terms of the ideal I. Using this framework, we demonstrate criteria for determining whether a ring is F-split or F-regular.