A Frobenius splitting on a ring R containing F_p is a map phi : R->R such that (1) it's additive (2) phi(a^p b) = a phi(b) (3) phi(1) = 1.
Def: a compatibly split ideal I is one such that phi(I) stays in I.
Thm:
0) Frobenius split rings are reduced (they have no nilpotents).
1) If I is compatibly split, then R/I is split.
2) ...and therefore I is radical.
3) If I,J are split then I+J and I intersect J are split.
4) If I is split, then I:J is split.
