4692

4419

كارشناسي ارشد

جبر

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1387

[پنج]،90،[II]ص.

ص.ع.به فارسي و انگليسي

محمود بهبودي

حسين خبازيان

25/7/88

شكرالله سالاريان، عاطفه قرباني

رياضي

ID4419

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

A Prime Ideal Prenciple in Commutative Algebra Rasool Zarei r zarei@math iut ac ir January 24 2008 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Mahmood Behboodi mbehbood@cc iut ac ir2000 MSC 11R44 13A15 Key words Commutative algebra Commutative rings Prime ideals Ideal families Prime ideal principles Module categories Abstract In this thesis we present an expanded account of Prime Ideal Principle based on an articleby T Y Lam and Manuel L Reyes 2008 for proving that certain ideals in a commutativering are prime This leads to a direct and uniform treatment of a number of standard resulton prime ideals in commutative algebra due to Krull Cohen Kaplansky Herstein Isaacs Mcdam D D Anderson and others More signi cantly the simple nature of the Prime IdealPrinciple enable us to generate a large number of hitherto unknown result of the maximalimplies prime variety The key notion used in our uniform approach to such prime idealproblems are those of Oka familise and Ako familise of ideals in a commutative ring de nein this thesis a clear statement of Oka result in the general setting of commutative ringsapparently rst appeared in Nagata S book Local Rings In this thesis we introducean elementary Prime Ideal Principle which states that for suitable ideal familie s F in acommutative ring every ideal maximal with respect to not being in F is prime This Principleopplies readily to retrieve all other result of the same kind in the literature that the authorsare aware of them In Section 3 we give applications of the Prime Ideal Principle by rst deriving uniformlyall known cases of the maximal implies prime results that we are aware of A rather pleasantfact here is that even D D Andersons theorem on minimal primes in An turned out to bejust a special case of the Prime Ideal Principle Various new cases of applications of thisPrinciple are then taken up in the second half of Section 3 LetMc R denote the category ofcyclic modules over a ring R In Sections 4 5 after setting up the correspondence betweenideal families in R and subcategories of Mc R we revisit the many types of ideal familiesintroduced in Section 2 and give categorical interpretations for the de ning properties ofsome of these families Most notably an Oka family of ideals in R is seen to correspondto a subcategory of Mc R that is closed under extensions With this categorical view ofOka families many examples of such families studied in Section 3 turn out to correspondto various familiar subcategories ofMc R that are clearly closed under extensions from themodule theoretic viewpoint 1

محمود بهبودي

حسين خبازيان

شكرالله سالاريان، عاطفه قرباني