Let (R,m) be a noetherian local ring of prime characteristic andFR(-) be the Frobenius functor. We will show that for all finiteR-modulesMthere is ann∈ N (depending onM) such that the fundamental theorem for modules over principal domains holds fori.e. there arer, s∈ N and elementsif and only ifRis geometrically unibranch in the sense of Grothendieck [2] and dimR≤ 1.For exampleRis geometrically unibranch, if it is complete andR/mis an algebraically closed field.