Convex Polytopes and Factorization Properties in Generalized Power Series Domains

Abstract

It is shown how to associate to any polytope that is not a simplex and any field K, a commutative integral domain D which has no irreducible elements and which is not pre-Schreier. The integral domain D is a generalized power series ring over K. Let R be an integral domain with quotient field K. Recall that a ∈ R {0} is said to be irreducible, or an atom, if a is not the product of two nonunits of R, and that a is said to be prime if, for all b, c ∈ R, a|bc implies a|b or a|c. It is easy to show that any prime element is irreducible, and much research has been done into the question of when the converse is true. For example, in any pre-Schreier domain, all irreducible elements are prime, and so we recall the definition: An element a of an integral domain R is primal if, whenever a divides bc with b and c in R, then a = b′c′ for some b′, c′ ∈ R where b′ divides b and c′ divides c. An integral domain in which each element is primal is said to be pre-Schreier. (A Schreier domain is a pre-Schreier domain which is also integrally closed.) Such rings have been studied by many authors, for example, [6], [7], [9], [13], [18], [22]. It is immediate that, in a pre-Schreier domain, each irreducible element is prime. On the other hand, there exist examples of integral domains which are not pre-Schreier, but in which each irreducible element is prime (see [18, Example 3.7]). (See also [1] for a comparison of these properties and several related ones.) In [21], W. C. Waterhouse shows that, if each quadratic polynomial f ∈ R[X] factors into linear polynomials in R[X] whenever it factors into linear polynomials in K[X], then every irreducible element in R is prime. The relation between this result and the pre-Schreier condition was explored in [18]