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Thesis (Ph.D.)--University of Nebraska-Lincoln, 2007. / Title from title screen (site viewed Apr. 29, 2008). PDF text: 103 p. : ill. ; 493 K. UMI publication number: AAT 3283908. Includes bibliographical references. Also available in microfilm and microfiche formats.

This paper is an investigation of several basic properties of noetherian rings. Chapter I gives a brief introduction, statements of definitions, and statements of theorems without proof. Some of the main results in the study of noetherian rings are proved in Chapter II. These results include proofs of the equivalence of the maximal condition, the ascending chain condition, and that every ideal is finitely generated. Some other results are that if a ring R is noetherian, then R[x] is noetherian, and that if every prime ideal of a ring R is finitely generated, then R is noetherian.

<p> P. Gabriel constructed rings of quotients by inverting elements of multiplicative sets which satisfy the Ore and the reversibility conditions. We employ this technique in our study of localizations of non-noetherian rings at Goldie semiprime ideals. The three types of clans developed in this thesis enable us to decompose in a unique fashion (weakly) classical sets of prime ideals into (weak) clans which, in essence, are minimal localizable sets of prime ideals, satisfying certain properties. We further show that these (weak) clans are mutually disjoint sets. The different types of rings, brought into consideration, exhibit many interesting properties in the context of our localization theory.</p> <p> We characterize the AR-property for the Jacobson radical of a semilocal ring by considering finitely generated modules. In the study of rings which are module-finite over their centres, we describe expressly the injective hull of the semilocal ring modulo its Jacobson radical. These two facts enable us to establish an interrelationship between the (strongly) classical semiprime ideals of the ring and those of its central subring. Furthermore, we show that under certain conditions the Q-sets are precisely all the minimal localizable sets of prime ideals of the ring. In the case of group rings, the flatness condition can be lifted without jeopardizing the validity of the assertion.</p> <p> Lastly, we apply localization technique to characterize the group theoretic notion of q-nilpotency.</p> / Thesis / Doctor of Philosophy (PhD)

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We say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a _nite 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 and in a 2011 paper Ay and Klingler obtain similar results for Noetherian reduced rings. We characterize the UDI property for Noetherian rings in general. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

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Thesis (Ph. D.)--University of Oregon, 2005. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 48-49). Also available for download via the World Wide Web; free to University of Oregon users.