ON LASKERIAN RINGS

Abstract

Abstract: This study explains that a strongly Laskerian domain D satisfies ACCP, every non zero ideal of D can be uniquely expressed as a product of primary ideals whose radicals are all distinct if D is one dimensional, D is completely integrally closed if it is a QR-domain. Furthermore if each overring (respectively valuation overring) of an integral domain D is strongly Laskerian then integral closure of D is a Dedekind domain (respectively an almost Dedekind domain).

[3] Title: A non-Noetherian Laskerian domain in which every Primary ideal is a valuation ideal vi

Abstract: The purpose of this study is to characterize Laskerian domains with the property that every primary ideal is a valuation ideal.

[4] Title: Factorization properties and chain conditions on ideals: A linkage Abstract: The purpose of this study is to find relationship among the various domains. In particular, the domains possessing factorization properties and the domains which hold different chain conditions on ideals.

[5] Title: Complete Integral Closure of a Domain in an Extension Field and Laskerian Valuation Domains Abstract: Suppose D is an integral domain with quotient field K, and assume that the complete integral closure of D is an intersection of Laskerian valuation domains on K. If L is an extension field of K, then the complete integral closure of D in L is an intersection of Laskerian valuation domains on L.

[6] Title: Polynomial and formal power series extensions of a strongly Laskerian domain and chain conditions Abstract: In this paper we study different chain conditions in polynomial and formal series extensions of a strongly Laskerian ring.

[7] Title: Fuzzy ideals in Laskerian rings Abstract: The aim of this paper is to introduce strongly primary fuzzy ideals and strongly irreducible fuzzy ideals in rings. We examine finite intersection property of some fuzzy ideals. That is finite intersection of prime fuzzy ideals, primary fuzzy ideals, irreducible fuzzy ideals and strongly irreducible fuzzy ideal is prime fuzzy ideal, primary fuzzy ideal, irreducible fuzzy ideal and strongly irreducible fuzzy ideal respectively. We discussed strongly irreducible fuzzy ideals in Laskerian rings. Moreover we showed that: A ring R is Laskerian if and only if Rµ is Laskerian for every fuzzy ideal µ of R.