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DERIVATIONS AND DIFFERENTIALS
Title
DERIVATIONS AND DIFFERENTIALS.
Creator
BASTIDA, VICKI C., Florida Atlantic University, Bastida, Julio R.
Abstract/Description
In this thesis we present a detailed study of the foundations of the general theory of derivations and differentials for commutative algebras over commutative rings. We give a direct and natural proof of the existence of the module of differentials, and then discuss some of its basic properties. A considerable part of the work is devoted to the establishing of the two fundamental exact sequences.
Date Issued
1971
PURL
http://purl.flvc.org/fcla/dt/13447
Subject Headings
Algebra, Abstract
Format
Document (PDF)
computation of the Hall coefficient g(q)[('6,4,2)(,42)(,4,2)]
Title
A computation of the Hall coefficient g(q)[('6,4,2)(,42)(,4,2)].
Creator
Anez, Myriam T., Florida Atlantic University, Schmidmeier, Markus
Abstract/Description
Let L be a uniserial ring of length n, with maximal ideal r , and finite residue field Λ/ r . We consider Λ-modules which possess a finite composition series. We note that a Λ-module has the form B ≅ ⨁i=1m Λ/ rli , where the type of B is the partition l = ( l1,&ldots;,lm ) denoted by t(B). For Λ-modules A, B, C with t(A) = m , t(B) = l , t(C) = n , if A ⊆ B, and B/A ≅ C, we define GBAC = |{U ⊆ B : U ≅ A and B/U ≅ C}|. We show that GBAC = MonoA,B,C Aut A = | S (A, B, C)/∼| = glmn (q),...
Show moreLet L be a uniserial ring of length n, with maximal ideal r , and finite residue field Λ/ r . We consider Λ-modules which possess a finite composition series. We note that a Λ-module has the form B ≅ ⨁i=1m Λ/ rli , where the type of B is the partition l = ( l1,&ldots;,lm ) denoted by t(B). For Λ-modules A, B, C with t(A) = m , t(B) = l , t(C) = n , if A ⊆ B, and B/A ≅ C, we define GBAC = |{U ⊆ B : U ≅ A and B/U ≅ C}|. We show that GBAC = MonoA,B,C Aut A = | S (A, B, C)/∼| = glmn (q), where |Λ/ r | = q, and the last equality comes from evaluating the Hall polynomial glmn (t) ∈ Z [t] at q, as stated in Hall's Theorem. We note that GBAC make up the coefficients of the Hall algebra. We provide a proof that the Hall algebra is a commutative and associative ring. Using the property of associativity of the Hall algebra and I. G. MacDonald's formula: glb1l =qnl -nb-n 1li≥ 1l'i -b'i,b' i-l'i+1 q-1 we develop a procedure to generate arbitrary Hall polynomials and we compute g6,4,2 4,24,2 (q).
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Date Issued
2005
PURL
http://purl.flvc.org/fcla/dt/13289
Subject Headings
Mathematical statistics, Algebra, Abstract, Abelian groups
Format
Document (PDF)
Unique decomposition of direct sums of ideals
Title
Unique decomposition of direct sums of ideals.
Creator
Ay, Basak., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
We say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In Chapters 1-3 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of one...
Show moreWe say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In Chapters 1-3 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of one-dimensional reduced commutative Noetherian rings with the UDI property have the UDI property, also. In Chapter 5 we show that the UDI property implies the Krull-Schmidt property for direct sums of torsion-free rank one modules for a reduced local commutative Noetherian one-dimensional ring R.
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Date Issued
2010
PURL
http://purl.flvc.org/FAU/2683133
Subject Headings
Algebraic number theory, Modules (Algebra), Noetherian rings, Commutative rings, Algebra, Abstract
Format
Document (PDF)
Minimal zero-dimensional extensions
Title
Minimal zero-dimensional extensions.
Creator
Chiorescu, Marcela, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
The structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are...
Show moreThe structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are ideals of R primary for distinct prime ideals of height greater than zero. A subsidiary result is the classification of minimal zero-dimensional extensions of general ZPI-rings.
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Date Issued
2009
PURL
http://purl.flvc.org/FAU/210447
Subject Headings
Algebra, Abstract, Noetherian rings, Commutative rings, Modules (Algebra), Algebraic number theory
Format
Document (PDF)
Algebraic and combinatorial aspects of group factorizations
Title
Algebraic and combinatorial aspects of group factorizations.
Creator
Bozovic, Vladimir., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
The aim of this work is to investigate some algebraic and combinatorial aspects of group factorizations. The main contribution of this dissertation is a set of new results regarding factorization of groups, with emphasis on the nonabelian case. We introduce a novel technique for factorization of groups, the so-called free mappings, a powerful tool for factorization of a wide class of abelian and non-abelian groups. By applying a certain group action on the blocks of a factorization, a number...
Show moreThe aim of this work is to investigate some algebraic and combinatorial aspects of group factorizations. The main contribution of this dissertation is a set of new results regarding factorization of groups, with emphasis on the nonabelian case. We introduce a novel technique for factorization of groups, the so-called free mappings, a powerful tool for factorization of a wide class of abelian and non-abelian groups. By applying a certain group action on the blocks of a factorization, a number of combinatorial and computational problems were noted and studied. In particular, we analyze the case of the group Aut(Zn) acting on blocks of factorization of Zn. We present new theoretical facts that reveal the numerical structure of the stabilizer of a set in Zn, under the action of Aut(Zn). New algorithms for finding the stabilizer of a set and checking whether two sets belong to the same orbit are proposed.
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Date Issued
2008
PURL
http://purl.flvc.org/FAU/107805
Subject Headings
Physical measurements, Mapping (Mathematics), Combinatorial enumeration problems, Algebra, Abstract
Format
Document (PDF)