The subject of this paper is the simultaneous ideal theory of a pair of integral domains R and G ≥ R, of which R is integrally closed, and G integrally dependent on R. It is assumed that the quotient field L of G is a finite separable extension of the quotient field K of R. The device of quotient rings effects a preliminary simplification in many of the proofs; the quotient rings R S and G S , with respect to any existent multiplicatively closed set S of non-zero elements of R, also satisfy the above basic postulates for R and G. Another method of preliminary simplification, valuable in the discussion of ramification theory, is the adjunction of Kronecker indeterminates. Such indeterminates (algebraically independent over K ) are denoted by y or z ; in connexion with the regular representation of L , they are regarded as adjoined to K .