## Monoidal Extensions of a Locally Quasi-Unmixed Unique Factorization Domain

- Author(s): Oeser, Paul Richard
- Advisor(s): Rush, David E
- et al.

## Abstract

Let *R* be a locally quasi-unmixed domain, *a, b _{1},&hellip, b_{n}* an

asymptotic sequence in *R*, *I*=(*a, b _{1},&hellip, b_{n}*)

*R*, and

*S*=*R*[*b _{1}/a,&hellip, b_{n}/a*]=

*R*[

*I/a*]. Then

*S*is a locally quasi-unmixed

domain, *a, b _{1}/a,&hellip, b_{n}/a* is an asymptotic sequence in

*S*, and

there is a one-to-one correspondence between the asymptotic primes

Ass_{R}(*R*/(*I*^{n})_{a}) of *I* and the asymptotic primes Ass_{S}(*S*/(*aS*^{n})_{a})

of *aS*=*IS*. Moreover, if *a, b _{1},&hellip, b_{n}* is an

*R*-sequence,

then that one-to-one correspondence extends between Ass_{R}(*R*/*I*) and

Ass_{S}(*S*/*aS*).

We give a sufficient condition for the monoidal transform

*S* to be a unique factorization domain, or a Krull domain whose class

group is torsion, finite, or finite cyclic. As a corollary, we give a

necessary and sufficient condition for *R* and its monoidal transform

to have the same class group.

In the case that *R* is a unique factorization domain, we examine the

height-one prime ideals of *S* to determine how far *S* is from unique

factorization. In Section 3.2, a complete description is given of which

height-one prime ideals *P* of *S* are principal or have a prinicpal primary

ideal in the case that the contraction to *R* of *P*, which we will call *p*, has height one. In Section 3.3, we show that if the

prime factors of *a* satisfy a mild condition, we may give a similar

description in the case that *p* has height greater than one. We give a necessary and

sufficient condition for *S* to be a Krull domain with finite cyclic

class group in the case that *a* is a power of a prime element, and we show

that this holds for the Rees ring *R*[*1/t,It*] as a monoidal transform

over *R*[*1/t*] as well. Furthermore, if *a* is a power of a prime element,

we show that if Rad(*I*) is not prime and *p* is a height-one prime ideal

of *R* contained in at least one but not all asymptotic prime divisors of *I*, then

the height-one prime ideal *P* of *S*, which can be represented as *pR*[*1/a*] intersected with *S*, has no principal primary

ideal.