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Extending the Skolem Property

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2017, Doctor of Philosophy, Ohio State University, Mathematics.
Skolem properties describe how well ideals of rings of integer-valued polynomials are characterized by their images under evaluation maps. They are usually defined only for finitely generated ideals. Evaluation is sensible for any ring made of polynomials, and it usually makes sense in the context of rational functions. We generalize the notion of a Skolem property to these broader settings. We give several examples of rings exhibiting these properties, and we extend many of the results about Skolem properties of rings of integer-valued polynomials to rings comprising polynomials. We examine rings in which the interesting values occur at only a finite collection of points. We demonstrate that such rings have the almost strong Skolem property, and we use that result to characterize when they are Prufer domains. We also consider n-generator properties in that setting. Making signi cant progress toward classifying the Skolem properties for all integrally closed rings of polynomials, we consider valuation domains on K(x), for some field K, contracted to K[x]. In this setting we characterize the almost Skolem property. We also extend the notion of a Skolem closure so that it is a semistar operation, and we demonstrate that it is more natural to consider the Skolem property as a property of star ideals rather than one of finitely generated ideals. We end with an application of this new perspective to the classical ring of integer-valued polynomials Int(Z), answering the open question: What is the largest class of ideals on which Int(Z) has the (strong) Skolem property?
K. Alan Loper (Advisor)
Ivo Herzog (Committee Member)
Cosmin Roman (Committee Member)
120 p.

Recommended Citations

Citations

  • Steward, M. (2017). Extending the Skolem Property [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1492517341492202

    APA Style (7th edition)

  • Steward, Michael. Extending the Skolem Property. 2017. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1492517341492202.

    MLA Style (8th edition)

  • Steward, Michael. "Extending the Skolem Property." Doctoral dissertation, Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1492517341492202

    Chicago Manual of Style (17th edition)