Publication Date

Fall 12-15-2024

Abstract

Inspired by the works of Petro, Epstein, Vassilev, and Morre, this thesis aims to study the generalized definitions of the semiprime operation, weakly prime operation, and standard closure operation on rings, that is, the multiplicative operation, weakly multiplicative operation, and standardly multiplicative operation on submodules. Then, we will use Matlis duality to induce the dual notions of these definitions on submodules of Matlis-dualizable Artinian modules. In order to understand the dual notion of the standardly multiplicative operation, that is, the standardly quotient operation, we will classify the operations on the injective hull of residue field of the ring K[[x,y]]/(xy) which are standardly quotient, idempotent, and order-preserving.

Degree Name

Mathematics

Level of Degree

Masters

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Janet Vassilev

Second Committee Member

Alexandru Buium

Third Committee Member

Hongnian Huang

Language

English

Keywords

Matlis duality, injective hull of residue field, interior operation, closure operation, multiplicative operation, quotient operation

Document Type

Thesis

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