Let R be a commutative ring with identity and let G and H be abelian groups with the group algebras RG and RH isomorphic as R-algebras. In this dissertation we investigate the relationships between G and H. Let inv(R) denote the set of rational prime numbers that are units in R and let G(R) (respectively, H(R)) be the direct sum of the p-components of G (respectively, H) with p ∈ inv(R). It is known that if G(R) or H(R) is nontrivial then it is not necessarily true that G and H are isomorphic. However, if R is an integral domain of characteristic 0 or a finitely generated indecomposable ring of characteristic 0 then G/G(R) ≅ H/H(R). This leads us to make the following definition: We say that R satisfies the Isomorphism Theorem if whenever RG ≅ RH as R-algebras for abelian groups G and H then G/G(R) ≅ H/H(R). Thus integral domains of characteristic 0 and finitely generated indecomposable rings of characteristic 0 satisfy the Isomorphism Theorem. Our first major result shows that indecomposable rings of characteristic 0 (no restrictions on generation) satisfy the Isomorphism theorem. It has been conjectured that all rings R satisfy the Isomorphism Theorem. However, we show that the conjecture may fail if nontrivial idempotents are present in R. This leads us to consider necessary and sufficient conditions for rings to satisfy the Isomorphism Theorem. We say that R is an ND-ring if whenever R is written as a finite product of rings then one of the factors, say Rᵢ, satisfies inv(Rᵢ) = inv(R). We show that every ring satisfying the Isomorphism Theorem is an ND-ring. Moreover, if R is an ND-ring and if inv(R) is not the complement of a single prime we show that R must satisfy the Isomorphism Theorem. This result together with some other fragmentary evidence leads us to conjecture that R satisfies the Isomorphism Theorem if and only if R is an ND-ring. Finally we obtain several equivalent formulations of our conjecture. Among them is the result that every ND-ring satisfies the Isomorphism Theorem if and only if every field of prime characteristic satisfies the Isomorphism Theorem.

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This thesis describes properties of Abelian groups, and develops a study of the properties of divisibility in Abelian groups.

The context for the development of this work is constructive mathematics without the axiom of countable choice. By constructive mathematics, we mean mathematics done without the law of excluded middle. Our original goal was to give a list of axioms for the real numbers R by only considering the order on R. We instead develop a theory of ordered sets and their completions and a theory of ordered abelian groups. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2018. / FAU Electronic Theses and Dissertations Collection

Thesis (M. Sc.)--University of Canterbury, 2009. / Typescript (photocopy). Includes bibliographical references (p. 47-48). Also available via the World Wide Web.

In this paper we prove that every element in the finite Abelian group Z(p) x Z(p) can be written as a sum over a subset of the set A, where A is any set of non-zero elements of Z(p) x Z(p) with / A / = 2p - 2.

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Thesis (Ph.D.)--Tufts University, 2001. / Advisers: Mauricio Gutierrez; Zbigniew Nitecki. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 73-74). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

Thesis (Ph.D.)--McMaster University (Canada), 1990. / Source: Dissertation Abstracts International, Volume: 52-10, Section: B, page: 5312. Supervisor: B. Banaschewski.

Thesis (Ph. D.)--Florida State University, 2004. / Advisor: Dr. Wolfgang Heil, Florida State University, College of Arts and Sciences, Dept. of Mathematics. Title and description from dissertation home page (viewed Sept. 28, 2004). Includes bibliographical references.