Two theorems on excellent rings

Approximation theorems for valuations on commutative rings

Chains of Unusual Excellent Local Rings

Let (T,M) be a complete local domain containing the integers. Let p1 ⊆ p2 ⊆ · · · ⊆ pn be a chain of nonmaximal prime ideals of T such that Tpn is a regular local ring. We construct a chain of excellent local domains An ⊆ An−1 ⊆ · · · ⊆ A1 such that for each 1 ≤ i ≤ n, the completion of Ai is T , the generic formal fiber of Ai is local with maximal ideal pi, and if I is a nonzero ideal of Ai th...

Structure theorems for AP rings

In “New Proofs of the structure theorems for Witt Rings”, the first author shows how the standard ring-theoretic results on the Witt ring can be deduced in a quick and elementary way from the fact that the Witt ring of a field is integral and from the specific nature of the explicit annihilating polynomials he provides. We will show in the present article that the same structure results hold fo...

COMMUTATIVlTY THEOREMS FOR RINGS WITH CONSTRAINTS ON COMMUTATORS

In this paper, we generalize sone well-known commutativity theorems for associative rings as follows: Let ’, > 1. ,,, .,, and be fixed nou-ncgative integers such that s ik m1, or i/= n1, and let R be a ring xvith unity satisfying the polynomial identity y*[x’,y] [x,y’]x for all y R. Sul,lose that (i) R has Q(z) (that is n[x,y] 0 implies [z,y] 0); (ii) the set of d] nilpotent ,,lem,’nts of R is ...

Two theorems on experimental logics

A generalization of a formal system is considered, in which the axioms of the formal system can be withdrawn or supplemented, as mechanical experimentation proceeds through time and the consequences of various combinations of assumptions are realized. The "theorems" of these experimental logics are taken to be those assertions possessing a proof which remains valid for all "sufficiently large t...