In this thesis, we are concerned with various aspects of fuzzy ideals of commutative rings. The central theorem is that of primary decomposition of a fuzzy ideal as an intersection of fuzzy primary ideals in a commutative Noetherian ring. We establish the existence and the two uniqueness theorems of primary decomposition of any fuzzy ideal with membership value 1 at the zero element. In proving this central result, we build up the necessary tools such as fuzzy primary ideals and the related concept of fuzzy maximal ideals, fuzzy prime ideals and fuzzy radicals. Another approach explores various characterizations of fuzzy ideals, namely, generation and level cuts of fuzzy ideals, relation between fuzzy ideals, congruences and quotient fuzzy rings. We also tie up several authors' seemingly different definitions of fuzzy prime, primary, semiprimary and fuzzy radicals available in the literature and show some of their equivalences and implications, providing counter-examples where certain implications fail.

Submitted by Automa??o e Estat?stica (sst@bczm.ufrn.br) on 2016-08-09T22:42:29Z No. of bitstreams: 1 FlaullesBooneBergamaschi_TESE.pdf: 610639 bytes, checksum: 58e4c11131b6241c88c80e6a1a2f579d (MD5) / Approved for entry into archive by Arlan Eloi Leite Silva (eloihistoriador@yahoo.com.br) on 2016-08-10T22:02:43Z (GMT) No. of bitstreams: 1 FlaullesBooneBergamaschi_TESE.pdf: 610639 bytes, checksum: 58e4c11131b6241c88c80e6a1a2f579d (MD5) / Made available in DSpace on 2016-08-10T22:02:43Z (GMT). No. of bitstreams: 1 FlaullesBooneBergamaschi_TESE.pdf: 610639 bytes, checksum: 58e4c11131b6241c88c80e6a1a2f579d (MD5) Previous issue date: 2015-12-11 / O principal objetivo deste trabalho ? propor a no??o de primalidade forte e uniformemente forte no ambiente fuzzy. Assim, ? apresentado um conceito de ideal uniformemente primo independente de alpha-cortes, mas de certa forma compat?vel. Tamb?m ? investigada a primalidade forte nos quantales, propondo uma defini??o de ideal primo e uniformemente primo coerente com o ambiente n?o-comutativo que serve de base para a investiga??o em fuzzy. Como ferramenta adicional, este trabalho introduz o conceito de t/m sistemas para o ambiente fuzzy sugerindo um caminho alternativo para lidar com primalidade. Por fim, ? desenvolvido uma vers?o fuzzy do teorema da correspond?ncia e no??es iniciais sobre o radical de um ideal fuzzy. / The main aim of this investigation is to propose the notion of uniform and strong primeness in fuzzy environment. First, it is proposed and investigated the concept of fuzzy strongly prime and fuzzy uniformly strongly prime ideal. As an additional tool, the concept of t/m systems for fuzzy environment gives an alternative way to deal with primeness in fuzzy. Second, a fuzzy version of correspondence theorem and the radical of a fuzzy ideal are proposed. Finally, it is proposed a new concept of prime ideal for Quantales which enable us to deal with primeness in a noncommutative setting.

ABSTRACT POSSIBILISTIC INTERPRETATION OF MISTUNING IN BLADED DISKS BY FUZZY ALGEBRA Karatas, Hamit &Ccedil / aglar M.S., Department of Mechanical Engineering Supervisor: Prof. Dr. H. Nevzat &Ouml / zg&uuml / ven Co-supervisor: Asst. Prof. Dr. Ender Cigeroglu September 2012, 103 pages This study aims to define the possibilistic interpretation of mistuning and examine the way of determining the worst case situations and assessing reliability value to that case by using possibilistic methods. Furthermore, in this study, benefits of using possibilistic interpretation of mistuning in comparison to probabilistic interpretation of mistuning are investigated. For the possibilistic analysis of mistuned structures, uncertain mistuning parameters are modeled as fuzzy variables possessing possibility distributions. In this study, alpha-cut representations of fuzzy numbers are used which makes fuzzy variables to be represented by interval numbers at each and every confidence level. The solution of fuzzy equations of motion is governed by fuzzy algebra methods. The bounds of the solution of the fuzzy equation of motion, i.e. fuzzy vibration responses of the mistuned structure, are determined by the extension principle of fuzzy functions. The performance of the method for possibilistic interpretation of mistuning is investigated by comparing it to the probabilistic methods both computational and accuracy wise. For the comparison study, two different optimization tools &ndash / genetic algorithm as the global optimization tool and constrained nonlinear minimization method as the gradient based optimization tool- are utilized in possibilistic analysis and they are compared to solutions of probabilistic methods resulted from Monte-Carlo method. The performances of all of the methods are tested on both a cyclically symmetric lumped parameter model and a realistic reduced order finite element model.

This study aims to define the possibilistic interpretation of mistuning and examine the way of determining the worst case situations and assessing reliability value to that case by using possibilistic methods. Furthermore, in this study, benefits of using possibilistic interpretation of mistuning in comparison to probabilistic interpretation of mistuning are investigated. For the possibilistic analysis of mistuned structures, uncertain mistuning parameters are modeled as fuzzy variables possessing possibility distributions. In this study, alpha-cut representations of fuzzy numbers are used which makes fuzzy variables to be represented by interval numbers at each and every confidence level. The solution of fuzzy equations of motion is governed by fuzzy algebra methods. The bounds of the solution of the fuzzy equation of motion, i.e. fuzzy vibration responses of the mistuned structure, are determined by the extension principle of fuzzy functions. The performance of the method for possibilistic interpretation of mistuning is investigated by comparing it to the probabilistic methods both computational and accuracy wise. For the comparison study, two different optimization tools &ndash / genetic algorithm as the global optimization tool and constrained nonlinear minimization method as the gradient based optimization tool- are utilized in possibilistic analysis and they are compared to solutions of probabilistic methods resulted from Monte-Carlo method. The performances of all of the methods are tested on both a cyclically symmetric lumped parameter model and a realistic reduced order finite element model.