Let G be an affine connected algebraic group acting regularly on an affine integral scheme X = Spec(R) not necessarily of finite type, over an algebraically closed field K with a quotient field Consider a G-rational twisted RG-module M having a non-trivial principal R-submodule Rz of  We will show: if Rz is G‑invariant, then so is Kz. Furthermore we study on the G-rationality of Kz in several cases.

algebraic group, semi-invariant, Krull domain.