Title of thesis: CLASSIFICATION OF PRIME IDEALS IN INTEGRAL GROUP ALGEBRAS OF FINITE ABELIAN GROUPS
نویسندگان
چکیده
Title of thesis: CLASSIFICATION OF PRIME IDEALS IN INTEGRAL GROUP ALGEBRAS OF FINITE ABELIAN GROUPS Heather Mallie McDonough, Master of Arts, 2005 Thesis directed by: Professor William Adams Department of Mathematics Let Z[G] be the integral group algebra of the group G. In this thesis, we consider the problem of determining all prime ideals of Z[G] where G is both finite and abelian. Because of Krull dimension arguments, there are only two types of prime ideals in Z[G]. First, we show that we can think of Z[G] as the quotient of a polynomial ring. Using this fact, and some Galois theory, we then classify the minimal prime ideals of our Z[G] where we restrict our group to having one or two generators. Next, we determine the form of the maximal ideals of Z[G] for the same case. However, the maximal ideals in our list need not be distinct. We further explore this issue restricting ourselves to cyclic groups. Using our previous work and cyclotomic field theory we are able to determine the duplication in our previous list. CLASSIFICATION OF PRIME IDEALS IN INTEGRAL GROUP ALGEBRAS OF FINITE ABELIAN GROUPS by Heather Mallie McDonough Thesis submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Master of Arts 2005 Advisory Committee: Professor William Adams, Chair/Advisor Professor Lawrence Washington Professor Michael Boyle c © Copyright by Heather McDonough 2005
منابع مشابه
Lubin-Tate Formal Groups and Local Class Field Theory
The goal of local class field theory is to classify abelian Galois extensions of a local field K. Several definitions of local fields are in use. In this thesis, local fields, which will be defined explicitly in Section 2, are fields that are complete with respect to a discrete valuation and have a finite residue field. A prototypical first example is Qp, the completion of Q with respect to the...
متن کاملNoetherian semigroup algebras and prime maximal orders
Let S be a semigroup and K be a field. A K-space K[S], with basis S and with multiplication extending, in a natural way, the operation on S, is called a semigroup algebra. It remains an open problem to characterize semigroup algebras that are a prime Noetherian maximal order. In this thesis, we give an answer to the problem for a large class of cancellative semigroups and we illustrate these re...
متن کامل2-recognizability of the simple groups $B_n(3)$ and $C_n(3)$ by prime graph
Let $G$ be a finite group and let $GK(G)$ be the prime graph of $G$. We assume that $ngeqslant 5 $ is an odd number. In this paper, we show that the simple groups $B_n(3)$ and $C_n(3)$ are 2-recognizable by their prime graphs. As consequences of the result, the characterizability of the groups $B_n(3)$ and $C_n(3)$ by their spectra and by the set of orders of maximal abelian subgroups are ...
متن کاملGraded Prime Ideals Attached to a Group Graded Module
Let $G$ be a finitely generated abelian group and $M$ be a $G$-graded $A$-module. In general, $G$-associated prime ideals to $M$ may not exist. In this paper, we introduce the concept of $G$-attached prime ideals to $M$ as a generalization of $G$-associated prime ideals which gives a connection between certain $G$-prime ideals and $G$-graded modules over a (not necessarily $G$-graded Noetherian...
متن کاملA Simple Classification of Finite Groups of Order p2q2
Suppose G is a group of order p^2q^2 where p>q are prime numbers and suppose P and Q are Sylow p-subgroups and Sylow q-subgroups of G, respectively. In this paper, we show that up to isomorphism, there are four groups of order p^2q^2 when Q and P are cyclic, three groups when Q is a cyclic and P is an elementary ablian group, p^2+3p/2+7 groups when Q is an elementary ablian group an...
متن کامل