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Commutative Hyperalgebra

Ramaruban, Nadesan
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1393237274

Abstract Details

2014, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
This work develops the notion of hyperrings as an extension of classical ring theory. Rings are sets that possess algebraic operations of addition and multiplication of elements in the set, encompassing examples such as the integers, rational numbers, etc. Hyperrings generalize rings further to allow multiple output values for the addition operation. In mathematics, introducing more abstract definitions often serves to clarify and consolidate phenomena that arise in a variety of settings. Hyperrings arise naturally in several settings in algebra, including quadratic form theory, number theory, orderings and ordered algebraic structures, tropical geometry, and multiplicative subgroups of fields. In this work we present an overview of the history of the development of the theory of hyperrings and other algebraic objects possessing hyperoperations. We continue by presenting a number of illustrative examples of hyperrings. We then determine the extent to which many results of classical commutative ring theory can be generalized to the setting of hyperrings. This in turn sheds light on the specific realizations of these objects. In particular, we consider homomorphisms of hyperrings, develop the hyperideal theory for hyperrings, introduce hypermodules and hypervector spaces, and examine the structure of specific classes of hyperrings such as Artinian and Noetherian hyperrings, Euclidean, principal hyperideal, and unique factorization hyperdomains, and hyperfields.
Tara Smith, Ph.D. (Committee Chair)
Timothy Hodges, Ph.D. (Committee Member)
James Osterburg, Ph.D. (Committee Member)
125 p.
Theoretical Mathematics
Hyperring; Hyperdomain; Hypermodule; Primary decomposition of a hyperideal; Noetherian hyperring; Artinian hyperring

Recommended Citations

Citations

  • Ramaruban, N. (2014). Commutative Hyperalgebra [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1393237274

    APA Style (7th edition)

  • Ramaruban, Nadesan. Commutative Hyperalgebra. 2014. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1393237274.

    MLA Style (8th edition)

  • Ramaruban, Nadesan. "Commutative Hyperalgebra." Doctoral dissertation, University of Cincinnati, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1393237274

    Chicago Manual of Style (17th edition)

ucin1393237274
1,263
© 2014, all rights reserved.
This open access ETD is published by University of Cincinnati and OhioLINK.