Commutative Ring Theory

16 Prime t-Ideals in R[X]

Evan G. Houston

Department of Mathematics

University of North Carolina at Charlotte

Charlotte, NC 28223 USA

Let R be an integral domain with quotient field K, and let X be an indeterminate. It is well known that (prime) t-ideals of R extend to (prime) t-ideals of R[X] and that every prime upper to zero $\text{fK}[\text{X}]\cap \text{R}[\text{X}]$ is a t-ideal of R[X]. We give a method for constructing prime t-ideals which assume neither of these forms; that is, we produce examples of prime t-ideals P in R[X] for which $0\ne (\text{P}\cap \text{R})[\text{X}]\ne \text{P}$. It is hoped that this construction will prove useful in investigating the t-dimension of polynomial rings.

1. Introduction and Preliminary Remarks

Let R be an integral domain with quotient field K. Recall that for a nonzero ...