The purpose of this note is to present new proofs of two classical theorems in ideal theory, both due to W. Krull: the intersection theorem (see Krull(1) for the original proof, and Northcott (2), Chapter III, where the theorem is stated in the form proved below), and the principal ideal theorem ((1), or (2), Chapter III, Theorem 6). Both proofs depend on a device introduced by the author in (3) in another connexion.

Let Λ denote the integral lattice in n-dimensional Euclidean space. A classical theorem of Minkowski's states that any bounded closed convex region K symmetrical in the origin 0 and with volume 2n contains a point of Λ other than 0. There will be a lattice point other than 0 in the interior of K except when K has certain forms, of which we will denote an arbitrary one by K*. An example of a K* is the cube |xi| ≤ 1 (i = 1, 2,..., n), and more generally a famous theorem of Hajós (3) states that if K* is a parallelepiped it is defined (except for integral unimodular transformations of the x's) by inequalities of the form |x1| ≤ 1, |a21x1 + x2| ≤ 1, …, |an1x1 + … + an, n-1xn-1+xn| ≤ 1.