This thesis investigates some of the properties of power series rings. The material is divided into three chapters. In Chapter I, some of the basic concepts of rings which are a prerequisite to an understanding of the definitions and theorems which follow are stated. Simple properties of power series rings are developed in Chapter II. Many properties of a ring R are preserved when we attach the indeterminant x to form the power series ring R[[x]]. Further results of power series rings are examined in Chapter III. An important result illustrated in this chapter is that power series rings possess some of the properties of rings of polynomials.

Thesis (Ph. D.)--University of Missouri-Columbia, 1998. / Typescript. Vita. Includes bibliographical references (leaf 93). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1998. / Typescript. Vita. Includes bibliographical references (leaf 93). Also available on the Internet.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2008. / Title from title screen (site viewed Sept. 18, 2008). PDF text: v, 109 p. : ill. ; 459 K. UMI publication number: AAT 3303652. Includes bibliographical references. Also available in microfilm and microfiche formats.

Let D[[X]] be the ring of formal power series over the commutative integral domain D. Gilmer has shown that if K is the quotient field of D, then D[[X]] and K[[X]] have the same quotient field if and only if K[[X]] ~ D[[X]]D_(O). Further, if a is any nonzero element of D, Sheldon has shown that either D[l/a][[X]] and D[[X]] have the same quotient field, or the quotient field of D[l/a][[X]] has infinite transcendence degree over the quotient field of D[[X]]. In this paper, the relationship between D[[X]] and J[[X]] is investigated for an arbitrary overring J of D. If D is integrally closed, it is shown that either J[[X]] and D[[X]] have the same quotient field, or the quotient field of J[[X]] has infinite transcendence degree over the quotient field of D[[X]]. It is shown further, that D is completely integrally closed if and only if the quotient field of J[[X]] has infinite transcendence degree over the quotient field of D[[X]] for each proper overring J of D. Several related results are given; for example, if D is Noetherian, and if J is a finite ring extension of D, then either J[[X]] and D[[X]] have the same quotient field or the quotient field of J[[X]] has infinite transcendence degree over the quotient field of D[[X]]. An example is given to show that if D is not integrally closed, J[[X]] may be algebraic over D[[X]] while J[[X]] and ~[[X]] have dif~erent quotient fields. / Ph. D.

Thesis (MSc (Mathematics))--University of Stellenboswch, 2010. / ENGLISH ABSTRACT: In the work of Green and Matignon it was shown that the Oort-Sekiguchi conjecture is equivalent to a local question of lifting automorphisms of power series rings. The Oort-Sekiguchi conjecture asks when an algebraic curve in characteristic p can be lifted to a relative curve in characteristic 0, while keeping the same automorphism group. The local formulation asks when an automorphism of a power series ring over a field k of characteristic p can be lifted to an automorphism of a power series ring over a discrete valuation ring with residue field k of the same order as the original automorphism. This thesis looks at the local formulation and surveys many of the results for this case. At the end it presents a new theorem giving a Hensel's Lemma type sufficient condition under which lifting is possible. / AFRIKAANSE OPSOMMING: Green en Matignon het bewys dat die Oort-Sekiguchi vermoede ekwivalent is aan `n lokale vraag oor of outomorfismes van magsreeksringe gelig kan word. Die Oort-Sekiguchi vermoede vra of `n algebra ese kromme in karakteristiek p gelig kan word na `n relatiewe kromme in karakteristiek 0, terwyl dit dieselfde outomorfisme groep behou. Die lokale vraag vra wanneer `n outomorfisme van `n magsreeksring oor `n liggaam k van karakteristiek p gelig kan word na `n outomorfisme van `n magsreeksring oor `n diskrete waarderingsring met residuliggaam k, terwyl dit dieselfde orde behou as die aanvanklike outomorfisme. Hierdie tesis fokus op die lokale vraag en bied `n opsomming van baie bekende resultate vir hierdie geval. Aan die einde word `n nuwe stelling aangebied wat voorwaardes stel waaronder hierdie vraag positief beantwoord kan word.

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