On the Structure and Ideal Theory of Complete Local Rings

Introduction. The concept of a local ring was introduced by Krull [7](1), who defined such a ring as a commutative ring 9î in which every ideal has a finite basis and in which the set m of all non-units is an ideal, necessarily maximal. He proved that the intersection of all the powers of m is the zero ideal. If the powers of m are introduced as a system of neighborhoods of zero, then 3Î thus becomes a topological ring, in which the usual topological notions —such as that of regular sequence (§2)—may be defined. The local ring dt is called complete if every regular sequence has a limit. Let m = (mi, «2, ■ ■ ■ , "*) and assume that no element of this basis may be omitted. If the ideals («i, u2, ■ ■ • ,Ui),i = l,2, ■ ■ • , n, are all prime, 9Î is said to be a regular local ring(2) of dimension ». It was conjectured by Krull [7, p. 219] that a complete regular local ring 9î of dimension ra whose characteristic equals that of its residue field 9î/tn is isomorphic to the ring of formal power series in ra variables with coefficients in this field. If on the other hand the characteristics are different, so that the characteristic of 9Î is zero and that of $K/m is a prime number p, then 9î cannot have this structure. In this case we note that p must be contained in m. Krull then conjectured that if 9î is unramified (that is, if p is not contained in m2), then 3Î is uniquely determined by its residue field and its dimension. He conjectured finally that every complete local ring is a homomorphic image of a complete regular local ring. These conjectures are proved in §§4—7, in particular, in Theorem 15 and its corollaries. Actually the basic result, to the proof of which is devoted Part II (§§4-6), is the one concerning arbitrary (that is, not necessarily regular) complete local rings. It is proved (Theorems 9 and 12) that every complete local ring 9Î is a homomorphic image of a complete regular local ring of a specific type, namely, the ring of all power series in a certain number of variables with coefficients taken from a field or from a valuation ring of a certain simple kind. This follows easily as soon as it is shown that a suitable "coefficient domain" can be imbedded in 9?, and the burden of the proof of