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Mirzadeh, Shahriar | 2011

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 42652 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Science
  6. Advisor(s): Mahdavi Hezavehi, Mohammad
  7. Abstract:
  8. In this Mater thesis, we o?er a general Prime Ideal Principle for proving that certain ideals in a commutative ring are prime. This leads to a direct and uniformtreatment of a number of standard results on prime ideals in commutative algebra,due to Krull, Cohen, Kaplansky, Herstein, Isaacs, McAdam, D.D. Anderson, andothers. More signi?cantly, the simple nature of this Prime Ideal Principle enablesus to generate a large number of hitherto unknown results of the “maximal impliesprime” variety. The key notions used in our uniform approach to such prime idealproblems are those of Oka families and Ako families of ideals in a commutative ring.In chapter 2, we amplify this study by developing further hierarchical relationshipsamong these families over special classes of rings, and also by ?nding concreteexamples of ideal families to show that certain such relationships cannot be reversedor improved. The special types of commutative rings studied here include vonNeumann regular rings and integral domains (for instance Dedekind domains andvaluation domains). In chapter 3, Completely prime right ideals are introduced asa one-sided generalization of the concept of a prime ideal in a commutative ring.Some of their basic properties are investigated, pointing out both similarities anddi?erences between these right ideals and their commutative counterparts
  9. Keywords:
  10. Prime Ideal ; Completely Prime Ideal ; Prime Ideal Principle ; Modules Category

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