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Lowest Terms in Commutative Rings.

Hasse, Erik Gregory.

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Lowest Terms in Commutative Rings.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Lowest Terms in Commutative Rings./
Author:	Hasse, Erik Gregory.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2018,
Description:	81 p.
Notes:	Source: Dissertation Abstracts International, Volume: 80-02(E), Section: B.
Contained By:	Dissertation Abstracts International80-02B(E).
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10826894
ISBN:	9780438382640

Lowest Terms in Commutative Rings.
Hasse, Erik Gregory.

Lowest Terms in Commutative Rings. - Ann Arbor : ProQuest Dissertations & Theses, 2018 - 81 p.

Source: Dissertation Abstracts International, Volume: 80-02(E), Section: B.

Thesis (Ph.D.)--The University of Iowa, 2018.

Putting fractions in lowest terms is a common problem for basic algebra courses, but it is rarely discussed in abstract algebra. In a 1990 paper, D.D. Anderson, D.F. Anderson, and M. Zafrullah published a paper called Factorization in Integral Domains, which summarized the results concerning different factorization properties in domains. In it, they defined an LT domain as one where every fraction is equal to a fraction in lowest terms. That is, for any x/y in the field of fractions of D, there is some a/b with x/y = a/b and the greatest common divisor of a and b is 1. In addition, R. Gilmer included a brief exercise concerning lowest terms over a domain in his book Multiplicative Ideal Theory. .

ISBN: 9780438382640Subjects--Topical Terms:

515831
Mathematics.

Lowest Terms in Commutative Rings.
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100 1 $a Hasse, Erik Gregory. $3 3429506
245 1 0 $a Lowest Terms in Commutative Rings.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2018
300 $a 81 p.
500 $a Source: Dissertation Abstracts International, Volume: 80-02(E), Section: B.
500 $a Adviser: Daniel D. Anderson.
502 $a Thesis (Ph.D.)--The University of Iowa, 2018.
520 $a Putting fractions in lowest terms is a common problem for basic algebra courses, but it is rarely discussed in abstract algebra. In a 1990 paper, D.D. Anderson, D.F. Anderson, and M. Zafrullah published a paper called Factorization in Integral Domains, which summarized the results concerning different factorization properties in domains. In it, they defined an LT domain as one where every fraction is equal to a fraction in lowest terms. That is, for any x/y in the field of fractions of D, there is some a/b with x/y = a/b and the greatest common divisor of a and b is 1. In addition, R. Gilmer included a brief exercise concerning lowest terms over a domain in his book Multiplicative Ideal Theory. .
520 $a In this thesis, we expand upon those definitions. First, in Chapter 2 we make a distinction between putting a fraction in lowest terms and reducing it to lowest terms. In the first case, we simply require the existence of an equal fraction which is in lowest terms, while the second requires an element which divides both the numerator and the denominator to reach lowest terms. We also define essentially unique lowest terms, which requires a fraction to have only one lowest terms representation up to unit multiples. We prove that a reduced lowest terms domain is equivalent to a weak GCD domain, and that a domain which is both a reduced lowest terms domain and a unique lowest terms domain is equivalent to a GCD domain. We also provide an example showing that not every domain is a lowest terms domain as well as an example showing that putting a fraction in lowest terms is a strictly weaker condition than reducing it to lowest terms.
520 $a Next, in Chapter 3 we discuss how lowest terms in a domain interacts with the polynomial ring. We prove that if D[T] is a unique lowest terms domain, then D must be a GCD domain. We also provide an alternative approach to some of the earlier results using the group of divisibility.
520 $a So far, all fractions have been representatives of the field of fractions of a domain. However, in Chapter 4 we examine fractions in other localizations of a domain. We define a necessary and sufficient condition on the multiplicatively closed set, and then examine how this relates to existing properties of multiplicatively closed sets.
520 $a Finally, in Chapter 5 we briefly examine lowest terms in rings with zero divisors. Because many properties of GCDs do not hold in such rings, this proved difficult. However, we were able to prove some results from Chapter 2 in this more general case.
590 $a School code: 0096.
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
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710 2 $a The University of Iowa. $b Mathematics. $3 1683492
773 0 $t Dissertation Abstracts International $g 80-02B(E).
790 $a 0096
791 $a Ph.D.
792 $a 2018
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10826894