## Çarpımsal Latisler

Institution Of The Thesis: Marmara University, Institute for Graduate Studies in Pure and Applied Sciences, Department of Mathematics, Turkey

Approval Date: 2012

Thesis Language: Turkish

Student: ZELİHA KILIÇ

Supervisor: Ünsal Tekir

Abstract:

In this study, principal elements in multiplicative lattices, localization in C-lattices, weak prime elements, weak ? -lattices, weak principal element lattices, weak r-lattices and regular lattices are investigated.Therefore, in the first and second section of the second chapter of this study, respectively, fundamental definitions and properties of multiplicative lattices and ring theory are given. Later on, after giving the definitions of both principal elements and C-lattices, various properties, lemmas and theorems of this type of elements and C-lattices, specially, the proof of prime avoidance theorem are given.In the third chapter, firstly, weak prime elements are defined and some theorems and results about weak prime elements are given. Meanwhile, some properties of the localization of weakly prime elements with respect to a specially defined multiplicatively closed subset are proven. Then almost prime elements are defined and some theorems familiar with the theorems for weak prime elements are given. Finally, by means of proving some properties of weak prime and weak principal elements, weak ? -lattices, weak principal element lattices, weak r-lattices and regular lattices are characterized. Moreover some theorems proved in this chapter are as follows:1)Let p be a weakly prime element of L. If p is not prime, then .2)Let . Then p is an almost prime element of L if and only if p is a weakly prime in .3)Let L be a weak r-lattice. Then the following statements are equivalent:(i) Every proper element of L is almost prime.(ii) Every proper principal element of L is almost prime.(iii) L is either a regular lattice or (L, m) is quasi-local with .