EAPSI: Extending the Scope of Mathematical Operations used as Tools to Study Rings

Mcgregor Daniel J, Columbus OH

Investigators

Abstract

A ring is a collection of elements that can be added, subtracted, or multiplied together, but not necessarily divided by one another. Rings appear in a wide range of areas within math and physics. The integers, the rational numbers, and polynomials are all commonly used examples of rings. A star operation is a useful tool for studying certain types of rings, particularly the properties of their multiplication. This project will extend the use of star operations to a larger category of rings than they are currently used for. This research will be conducted at Pohang University of Science and Technology in collaboration with Dr. Byung Gyun Kang, an expert in the field of commutative ring theory. Star operations are a closure operation on the ideals of a commutative ring satisfying certain axioms related to principal ideals. Commonly studied star operations are the b, t, and v operations, and they are an active area of research in multiplicative ideal theory. Star operations can also be used to construct Kronecker function rings, which are invaluable for studying valuation overrings and the topology of Zariski-Riemann spaces. They are mainly studied in the context of integral domains, and comparatively little work has been done in the case of rings with zero divisors. This project will expand the use of star operations to commutative rings with zero divisors, with an emphasis on generalizing Kronecker function rings and Zariski-Riemann spaces to that setting. This award under the East Asia and Pacific Summer Institutes program supports summer research by a U.S. graduate student and is jointly funded by NSF and the National Research Foundation of Korea.