Locally factorial integral domains

Every divisor class of Krull monoid domains contains a prime ideal

On the divisor class group of Rees-algebras

Let A be a Krull domain having infinitely many height one primes. It is shown that any ideal of height two in the polynomial ring A[t] contains a prime element. An application to the construction of Dedekind domains with specified class groups is given, along with an example to show the necessity of assuming infinitely many height one primes. Introduction. Krull domains can be viewed as generalizations of factorial rings and, inversely, factorial rings can be characterized as Krull domains with trivial divisor class group. As is well known, an integral domain is factorial if and only if every one of its nonzero prime ideals contains a nonzero principal prime ideal, i.e. a prime element. Thus, to assert the factoriality of an integral domain is to make a statement about the distribution of its prime elements. Factorial rings are those in which the prime elements are in some sense widely distributed. The purpose of this note is to present a result on the distribution of prime elements in integral domains of the form BA[t], where A is a Krull domain and t is an indeterminate. We shall prove that if A has infinitely many height one prime ideals, then every ideal in B of grade at least two contains a prime element. This theorem provides, for polynomial rings over Krull domains, a converse to the obvious fact that in any integral domain an ideal which properly contains a principal prime ideal has grade at least two. We shall give an example to show the necessity of the assumption that A has infinitely many height one prime ideals. Note that this assumption is equivalent to the assertion that A is not a semilocal principal ideal domain.

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On the distribution of prime elements in polynomial Krull domains

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On the distribution of prime elements in polynomial Krull domains