No description available.

No description available.

No description available.

No description available.

No description available.

No description available.

Thesis (Ph.D.)--York University, 2007. Graduate Programme in Mathematics. / Typescript. Includes bibliographical references (leaves 113-116). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR29337

Thesis (Ph. D.)--University of Oregon, 2005. / Typescript. Includes vita and abstract. Includes bibliographical references (leaf 152). Also available for download via the World Wide Web; free to University of Oregon users.

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.

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, August 2003. / Includes bibliographical references. Also available on the Internet.