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Let G be an affine algebraic group with an algebraic torus G over an algebraically closed field K of an arbitrary characteristic p. We show a criterion for G to be a finite central extension of G in terms of invariant theory of all regular actions of any closed subgroup H containing ZG(G ) on affine Krull K-schemes such that invariant rational functions are locally fractions of invariant regular functions. Consider an affine Krull H-scheme X = Spec(R) and a prime ideal P of R with ht(P) = ht(P ∩ R) = 1. Let I(P) denote the inertia group of P under the action of H . The group G is central over G if and only if the fraction e(P,P ∩ R)/e(P,P ∩ R) of ramification indices is equal to 1 (p = 0) or to the p-part of the order of the group of weights of G on R vanishing on R/P ∩R (p > 0) for an arbitrary X and P. MSC: primary 13A50, 14R20, 20G05; secondary 14L30, 14M25

VALUATIVE CHARACTERIZATION OF CENTRAL EXTENSIONS OF ALGEBRAIC TORI ON KRULL DOMAINS