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Abstract: Let G be a finite group acting on a polynomial ring A over the field K and let denote the corresponding ring of invariants. Let B be the subalgebra of generated by all homogeneous elements of degree less than or equal to the group order . Then in general B is not equal to if the characteristic of K divides . However we prove that the field of fractions coincides with the field of invariants . We also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for . We prove that there is always a nonzero transfer of degree , such that the localization can be generated by fractions of homogeneous invariants of degrees less than . If with finite-dimensional -module V, then c can be chosen in degree one and can be replaced by . Let denote the image of the classical Noether-homomorphism (see the definition in the paper). We prove that contains the transfer ideal and thus can be used to calculate generators for by standard elimination techniques using Gröbner-bases. This provides a new construction algorithm for . [Copyright &y& Elsevier]