Prime Properties of Irreducible Elements in a Birational Extension of A Noetherian UFD
by
Aihua Li
Loyola University New Orleans

Let R be a Noetherian UFD. Let B be a finitely generated birational extension of R; that is, B is a Noetherian integral domain between R and the quotient field of R. A finitely generated birational extension of a Noetherian UFD R has the form: R[g₁/f₁, g₂/f₂, ... , g_n/f_n], where f_i, g_i are nonzero elements of R for each i, and f_i and g_i are relatively prime. A birational extension of a UFD may not be a UFD. In this paper, we focus on the case when n=1, i.e., the birational extensions of type B=R[g/f], where R is a UFD. It is well known that a Noetherian integral domain is a UFD if and only if every irreducible element of it generates a (principal) prime ideal. This property fails in some birational extensions of R. If B=R[g/f] as above, it is natural to ask: ``In case B is not a UFD, what elements of B generate a prime ideal?'' Furthermore, if an irreducible element does not generate a prime ideal, we may ask, "Does it generate an ideal whose radical is prime?" Toward answering these questions, we give a class of irreducible elements of B which generate prime ideals. Furthermore, we prove that certain irreducible elements may generate non-prime ideals whose radicals are prime. Finally, A set of irreducible elements which do not generate ideals with prime radicals are also given.

Date received: November 15, 1998