Skip to Main Content
 

Global Search Box

 
 
 
 

Files

File List


Liu Dissertation.pdf (957.65 KB)

ETD Abstract Container

Abstract Header

Rings of Integer-Valued Rational Functions

Liu, Baian
http://rave.ohiolink.edu/etdc/view?acc_num=osu1681513836976336

Abstract Details

2023, Doctor of Philosophy, Ohio State University, Mathematics.
As objects that appear throughout mathematics, integer-valued polynomials have been studied extensively. However, integer-valued rational functions are a much less studied generalization. We consider the set of integer-valued rational functions over an integral domain as a ring and study the ring-theoretic properties of such rings. We explore when rings of integer-valued rational functions are Bézout domains, Prüfer domains, and globalized pseudovaluation domains. We completely classify when the ring of integer-valued rational functions over a valuation domain is a Prüfer domain and when it is a Bézout domain. We extend the classification of when rings of integer-valued rational functions are Prüfer domains. This includes a family of rings of integer-valued rational functions that are Prüfer domains, as well as a family of integer-valued rational functions that are not Prüfer domains. We determine that the classification of when rings of integer-valued rational functions are Prüfer domains is not analogous to the interpolation domain classification of when rings of integer-valued polynomials are Prüfer domains. We also show some conditions under which the ring of integer-valued rational functions is a globalized pseudovaluation domain. We also prove that even if a pseudovaluation domain has an associated valuation domain over which the ring of integer-valued rational functions is a Prüfer domain, the ring of integer-valued rational functions over the pseudovaluation domain is not guaranteed to be a globalized pseudovaluation domain. Because rings of integer-valued rational functions are rings of functions, we can study their properties with respect to evaluation. These properties include the Skolem property and its generalizations, which are properties concerning when ideals are able to be distinguished using evaluation. We connect the Skolem property to the maximal spectrum of a ring of integer-valued rational functions. This is then generalized using star operations. Another way to analyze the ring of integer-valued rational functions is through factorization theory. Unfortunately, it seems empirically that irreducible elements often do not exist in rings of integer-valued rational functions. We provide a family of rings whose ring of integer-valued rational functions is atomic and we analyze factorization invariants such as the set of factorization lengths and catenary degrees and compare these to those of the base ring. Lastly, we introduce the rational closure. This captures the idea that the set of rational functions that are integer-valued on two different sets can be the same. We show that if the base ring is a pseudovaluation domain, then the rational closure gives rise to a topology on the field of fractions that is equivalent to the topology induced by the associated valuation.
K. Alan Loper (Advisor)
Cosmin Roman (Committee Member)
Ivo Herzog (Committee Member)
176 p.
Mathematics
integer-valued rational function; multiplicative ideal theory; Prüfer domain; valuation domain; PVD; GPVD; commutative algebra; algebra; ring; Skolem property; factorization; non-D-ring; star operation; ultrafilter

Recommended Citations

Citations

  • Liu, B. (2023). Rings of Integer-Valued Rational Functions [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1681513836976336

    APA Style (7th edition)

  • Liu, Baian. Rings of Integer-Valued Rational Functions. 2023. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1681513836976336.

    MLA Style (8th edition)

  • Liu, Baian. "Rings of Integer-Valued Rational Functions." Doctoral dissertation, Ohio State University, 2023. http://rave.ohiolink.edu/etdc/view?acc_num=osu1681513836976336

    Chicago Manual of Style (17th edition)

osu1681513836976336
76
© 2023, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.