Let k be an algebraically closed field and G a linear algebraic group over k acting rationally on a k-algebra V. Generalizing work of Moeglin and Rentschler in characteristic zero, we study the action of G on the spectrum of rational ideals of V. The main result is the following. Suppose that V is semiprime left Goldie. Let L be a G-stable commutative semisimple subalgebra of the total ring of fractions Q(V) of V such that LG = k · 1L. This occurs, for example, if the zero ideal of V is G-rational and L is the center of Q(V). Then there is, for some closed subgroup H of G, a G-equivariant embedding v of L into Q(G/H) (the algebra of rational functions on G/H) such that Q(G/H) is purely inseparable over v(L). This has applications to the closure of the orbit of a rational ideal.