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On the Structure of Kronecker Function Rings and Their Generalizations

McGregor, Daniel
http://rave.ohiolink.edu/etdc/view?acc_num=osu1523924670762482

Abstract Details

2018, Doctor of Philosophy, Ohio State University, Mathematics.
Given a domain D with quotient field K, a Kronecker function ring E of D is an overring of some polynomial extension D[X] such that E intersect K = D and f/g in E for any nonzero f,g in D[X] with c(f) contained in c(g). These domains have many interesting properties which are already well-known. For example, Kronecker function rings are Bezout domains and they are associated with star operations on D and with collections of valuation overrings of D. Previous study on this topic has mainly focused on individual Kronecker function rings, particularly the ring Kr(D, b) associated with the b operation. In this thesis we will look at two generalizations of the concept of Kronecker function rings. One of these is Kr(R,*), which extends the definition beyond a domain D to a ring R with zero divisors. The other generalization we will investigate is the "Kronecker power series ring." This will involve overrings of D[[x]] rather than D[X]. We will see that Kronecker power series rings exhibit many of the same properties that Kronecker function rings do. There has already been some work demonstrating the existence of Kronecker function rings of R and Kronecker power series rings of D in some special cases. We expand on this by showing the necessary and sufficient conditions on a ring R which guarantee the existence of Kronecker function rings, and likewise for a domain D and Kronecker power series rings. We also move beyond the study of individual Kronecker function rings and examine Kr(D), the collection of all Kronecker function rings of a given domain D. One notable result we have is that Kr(D) is a spectral space under its Zariski topology. We define the concept of Kronecker dimension, which is connected to the space of valuation overrings of D. We also construct some nice examples demonstrating that Kronecker dimension is a nontrivial concept, distinct from both the Krull and valuative dimensions.
Alan Loper, PhD (Advisor)
Tariq Rizvi, PhD (Committee Member)
Cosmin Roman, PhD (Committee Member)
117 p.
Mathematics
Kronecker function rings; Kronecker power series rings; Kronecker dimension; star operations; valuations

Recommended Citations

Citations

  • McGregor, D. (2018). On the Structure of Kronecker Function Rings and Their Generalizations [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1523924670762482

    APA Style (7th edition)

  • McGregor, Daniel. On the Structure of Kronecker Function Rings and Their Generalizations. 2018. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1523924670762482.

    MLA Style (8th edition)

  • McGregor, Daniel. "On the Structure of Kronecker Function Rings and Their Generalizations." Doctoral dissertation, Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1523924670762482

    Chicago Manual of Style (17th edition)

osu1523924670762482
395
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