World Applied Sciences Journal 24 (1): 53-57, 2013 ISSN 1818-4952 © IDOSI Publications, 2013 DOI: 10.5829/idosi.wasj.2013.24.01.1611 Corresponding Author: A. Molkhasi, Department of Mathematical Sciences, University For Teachers, Tabriz, Iran. 53 Noetherian Topological Spaces and Cohen-Macaulay A. Molkhasi Department of Mathematical Sciences, University For Teachers, Iran-Tabriz, Institute of Mathematics and Mechanics Academy of Sciences of, Azerbaijan-Baku Submitted: Apr 25, 2013; Accepted: Jun 13, 2013; Published: Aug 07, 2013 Abstract: In this paper, R is commutative ring with identity and L is K-lattice. We prove that, if p Spec (L) and each A L is p-radical finite, then p is sequential Noetherian topological space and is s-compact. Furthermore, it has only a finite number of distinct irreducible components. Also, it is shown that if the lattice of ideals R is a principal lattice, then the prime spectrum Spec(R) is a sequential Noetherian topological space. Finally, it is shown that, if L(R) is a principal lattice and be the compact open of the Zariski spectrum of R, then R[ ] is Cohen-Macaulay ring. 2000 Mathematics Subject Classification: 54D55 54D20 06F05 13A15 Key words: Principal lattice Noetherian topological space Compact Sequential space Cohen-Macaulay ring INTRODUCTION studied by Gotchev. A topological space X is s-compact The concept of Noetherian topological spaces arises subcover. In section 2, we present some needed results naturally in the study of Noetherian rings and is of great on multiplicative lattices with ACC on radical elements. interest in some areas of mathematics such as Algebraic A topological space X is irreducible if X is non-empty, Geometry. A topological space (X, ) is called Noetherian and if any two non-empty open subsets of X intersect. if satisfies the ascending chain condition (ACC for In section 3, it shown that if p Spec(L) and each A L is short) : every strictly ascending chain U U ... of P-radically finite, then P is a sequential Noetherian 1 2 elements of is finite [1-4]. Topological spaces that topological space, is s- compact and it has only a finite satisfy properties similar to ACC have been widely number of distinct irreducible components. In section 4, studied. In [5], spaces with Noetherian bases have been it is pvoved that if L(R) is the lattice of ideals of a introduced (a topological space has a Noetherian base if commutative ring R with identity, then an ideal A of R is it has a base that satisfies ACC) and many interesting principal as member of L(R) if and only if A is finitely results about such spaces have been obtained [5-7]. generated and ARP is principal for each maximal ideal P of Clearly, every Noetherian space has a Noetherian base R. Also, we show that if L(R) is a principal lattice, then the but the converse is not true in general. An element p in prime spectrum Spec(R) is a sequential Noetherian the K-lattice L is said to be prime if ab p implies a p or topological space. Inparticular, it is shown that if L(R) is b p. Let Spec(L) denote the set of prime elements of L, a principal lattice and be the compact open of the which we give the Zariski topology. Zariski spectrum of R, then R[ ] is Cohen-Macaulay. If p Spec(L), we always give p the relative topology induced from the Zariski topology on Spec(L). Ascending Chain Condition on Radical Elements: If A L, we define the p-radical of A to be p-rad We recall the de definitions. We follow the terminology (A) = ^ { P| A } and call A a p-radical element if of [8, 9]. Let (L, ) be a complete lattice with maximal A = P-rad(A). If p Spec(L), we say that an element A of element R and minimal element 0=0 . Then L is said to L is P-radically finite if there exists a compact element F A be multiplicative lattice if L is a multiplicative ordered such that p-rad(F)=P-rad(A). In this paper, we study monoid such that the multiplication on L distributes over sequential properties of Noetherian topological spaces. arbitrary joins and such that R is the identity for the The concept of s-compactness was introduced and multiplication. if every sequentially open cover of X has a finite L