Mathematics, Department of
Department of Mathematics: Dissertations, Theses, and Student Research
Date of this Version
Summer 8-2012
Document Type
Thesis
Citation
A dissertation presented to the faculty of the Graduate College at the University of Nebraska in partial fulfillment of requirements for the degree of Doctor of Philosophy
Major: Mathematics
Under the supervision of Professor Thomas Marley
Lincoln, Nebraska, August 2012
Abstract
Rings graded by Z and Zd play a central role in algebraic geometry and commutative algebra, and the purpose of this thesis is to consider rings graded by any abelian group. A commutative ring is graded by an abelian group if the ring has a direct sum decomposition by additive subgroups of the ring indexed over the group, with the additional condition that multiplication in the ring is compatible with the group operation. In this thesis, we develop a theory of graded rings by defining analogues of familiar properties---such as chain conditions, dimension, and Cohen-Macaulayness. We then study the preservation of these properties when passing to gradings induced by quotients of the grading group.
Advisor: Thomas Marley
Comments
Copyright (c) 2012, Brian P. Johnson. Used by permission