aa r X i v : . [ m a t h . R A ] J u l On the prime spectrum of an le-module
M. Kumbhakar and A. K. Bhuniya
Department of Mathematics, Nistarini College, Purulia-723101, W.B.Department of Mathematics, Visva-Bharati, Santiniketan-731235, [email protected], [email protected]
Abstract
Here we continue to characterize a recently introduced notion, le-modules R M over a com-mutative ring R with unity [7]. This article introduces and characterizes Zariski topology on theset Spec ( M ) of all prime submodule elements of M . Thus we extend many results on Zariskitopology for modules over a ring to le-modules. The topological space Spec(M) is connected ifand only if R/Ann ( M ) contains no idempotents other than 0 and 1. Open sets in the Zariskitopology for the quotient ring R/Ann ( M ) induces a base of quasi-compact open sets for theZariski-topology on Spec(M). Every irreducible closed subset of Spec(M) has a generic point.Besides, we prove a number of different equivalent characterizations for Spec(M) to be spectral. Keywords:
Rings; complete lattice; le-modules; prime spectrum; Zariski topology.
AMS Subject Classifications:
W. Krull [20] recognized that many properties on ideals in a commutative ring are independent on thefact that they are composed of elements. Hence those properties can be restated considering idealsto be an undivided entity or element of a suitable algebraic system. In the abstract ideal theory,the ideals are considered to be elements of a multiplicative lattice, a lattice with a commutativemultiplication and satisfies some axioms. Ward and Dilworth [10], [11], [30], [31], [32], [33], [34],contributed many significant results in abstract ideal theory via the residuation operation on amultiplicative lattice. In [11] Dilworth redefined principal elements and obtained the Noether primary1ecomposition theorems. The multiplicative theory of ideals were continued towards dimensiontheory, representations problem and many other overlaping areas by D. D. Anderson, E. W. Johnson,J. A. Johnson, J. P. Ladiaev, K. P. Bogart, C. Jayaram and many others. For the proper referencesof these articles we refer to [3]. Also we refer to [21] for more discussions on abstract ideal theory.Success achieved in abstract ideal theory naturally motivated researchers to consider abstractsubmodule theory, which today is known as the theory of lattice modules. E. W. Johnson and J. A.Johnson [15], [18], introduced and studied Noetherian lattice modules. They also considered latticemodules over semilocal Noetherian lattice. Whitman [35] introduced principal elements in a latticemodule and extended Nagata’s principle of idealization to lattice modules. Nakkar and Anderson[28] studied localization in lattice modules. There are many articles devoted to lattice modules, tomention a few [27], [19], [16], [17], [35].This article is a continuation of our present project on le-modules, an algebraic structure moti-vated by lattice modules over a multiplicative lattice. Our goal is to develop an “abstract submoduletheory” which will be capable to give insight about rings more directly. The system we choose is acomplete lattice M with a commutative and associative addition which is completely join distributiveand admits a module like left action of a commutative ring R with 1. Since we are taking left actionof a ring R not of the complete modular lattice of all ideals of R , we hope that influence of arithmeticof R on M will be easier to understand. For further details and motivation for introducing le-moduleswe refer to [7].In this article we introduce and study Zariski topology on the set Spec( M ) of all prime submoduleelements of an le-module R M . It is well established that the Zariski topology on prime spectrumis a very efficient tool to give geometric interpretation of the arithmetic in rings [2], [14], [25], [29],[36] and modules [1], [5], [6], [12], [13], [22], [23], [24], [26]. Here we have extended several results onZariski topology in modules to le-modules.In addition to this introduction, this article comprises six sections. In Section 2, we recalldefinition of le-modules and various associated concepts from [7]. Also we discuss briefly on theZariski topology in rings and modules. Section 3, introduces Zariski topology τ ∗ ( M ) on the setSpec( M ) of all prime submodule elements of an le-module R M , and proves the equivalence of adifferent characterization of τ ∗ ( M ) by the ideals of the ring R . Annihilator Ann( M ) of an le-module R M is defined so that it is an ideal of R . This induces a natural connection between Spec( M )and Spec( R/Ann ( M )). Section 4 establishes some basic properties of this natural connection. As aconsequence, we deduce that Spec ( M ) is connected if and only if 0 and 1 are the only idempotents2f the ring R/Ann ( M ). Section 5 finds a base for the Zariski topology on Spec( M ) consisting ofquasi-compact open sets. Section 6 characterizes irreducible closed subsets and generic points inSpec( M ) showing that these two notions are closely related. Section 7 proves the equivalence of anumber of different characterizations for Spec(M) to be spectral. Throughout the article, R stands for a commutative ring with 1. The cardinality of a set X will bedenoted by | X | .First we recall the definition of an le-module and various associated concepts from [7]. Here byan le-semigroup we mean ( M, + , , e ) such that ( M, ≤ ) is a complete lattice, ( M, +) is a commutativemonoid with the zero element 0 M and for all m, m i ∈ M, i ∈ I it satisfies m ≤ e and(S) m + ( ∨ i ∈ I m i ) = ∨ i ∈ I ( m + m i ).Let R be a ring and ( M, + , , e ) be an le-semigroup. If there is a mapping R × M −→ M whichsatisfies(M1) r ( m + m ) = rm + rm ;(M2) ( r + r ) m r m + r m ;(M3) ( r r ) m = r ( r m );(M4) 1 R m = m ; 0 R m = r M = 0 M ;(M5) r ( ∨ i ∈ I m i ) = ∨ i ∈ I rm i ,for all r, r , r ∈ R and m, m , m , m i ∈ M, i ∈ I then M is called an le-module over R . It is denotedby R M or by M if it is not necessary to mention the ring R .From (M5), we have,(M5) ′ m m ⇒ rm rm , for all r ∈ R and m , m ∈ M .An element n of an le-module R M is said to be a submodule element if n + n, rn n , for all r ∈ R . We call a submodule element n proper if n = e . Note that 0 M = 0 R .n n , for everysubmodule element n of M . Also n + n = n , i.e. every submodule element of M is an idempotent.We define the sum of the family { n i } i ∈ I of submodule elements in R M by:3 i ∈ I n i = ∨{ ( n i + n i + · · · + n i k ) : k ∈ N , and i , i , · · · , i k ∈ I } .Since R M is assumed to be complete, P i ∈ I n i is well defined. It is easy to check that P i ∈ I n i is asubmodule element of M .If n is a submodule element in R M , then we denote( n : e ) = { r ∈ R : re n } .Then 0 R ∈ ( n : e ) implies that ( n : e ) = ∅ . One can check that ( n : e ) is an ideal of R . For submoduleelements n l of an le-module R M , we have ( n : e ) ⊆ ( l : e ). Also if { n i } i ∈ I be an arbitrary familyof submodule elements in R M , then ( ∧ i ∈ I n i : e ) = ∩ i ∈ I ( n i : e ).We call (0 M : e ) the annihilator of R M . It is denoted by Ann ( M ). Thus Ann ( M ) = { r ∈ R : re M } = { r ∈ R : re = 0 M } . For an ideal I of R , we define Ie = ∨{ P ki =1 a i e : k ∈ N ; a , a , · · · , a k ∈ I } Then Ie is a submodule element of M . Also for any two ideals I and J of R , I ⊆ J implies that Ie J e . The following result, proved in [7], is useful here.
Lemma 2.1.
Let R M be an le-module and n be a submodule element of M . Then for any ideal I of R , Ie n if and only if I ⊆ ( n : e ) . Now we recall some notions from rings and modules. An ideal P in a ring R is called prime iffor every a, b ∈ R , ab ∈ P implies that a ∈ P or b ∈ P . We denote the set of all prime ideals of aring R by X R or Spec( R ). For every ideal I of R , we define V R ( I ) = { P ∈ Spec ( R ) : I ⊆ P } , andτ ( R ) = { X R \ V R ( I ) : I is an ideal of R } . Then τ ( R ) is a topology on Spec( R ), which is known as the Zariski topology on Spec( R ). Thereare many enlightening characterizations associating arithmetical properties of R and topologicalproperties of Spec( R ) [29].Let M be a left R -module. Then a proper submodule P of M is called a prime submodule if forevery for r ∈ R and n ∈ M , rn ∈ P implies that either n ∈ P or rM ⊆ P . We denote the set of allprime submodules of M by Spec( M ). 4or a submodule N of M , ( N : M ) = { r ∈ R : rM ⊆ N } is an ideal of R . There is a topology τ ( M ) on Spec( M ) such that the closed subsets are of the form V ( N ) = { P ∈ Spec ( M ) : ( N : M ) ⊆ ( P : M ) } .The topology τ ( M ) is called the Zariski topology on M . Associating arithmetic of a module over aring R with the geometry of the Zariski topology on M is an active area of research on modules [1],[6], [12], [13], [22], [23], [24], [26].The notion of prime submodule elements was introduced in [7], which extends prime submodulesof a module over a ring. A proper submodule element p of an le-module R M is said to be a primesubmodule element if for every r ∈ R and n ∈ M , rn p implies that r ∈ ( p : e ) or n p . The prime spectrum of M is the set of all prime submodule elements of M and it is denoted by Spec( M )or X M . For P ∈ Spec ( R ), we denote Spec P ( M ) = { p ∈ Spec ( M ) : ( p : e ) = P } .We also have the following relation between prime submodule elements of an le-module R M andprime ideals of R . Lemma 2.2. [7] If p is a prime submodule element of R M , then ( p : e ) is a prime ideal of R . Also we refer to [4], [8] for background on commutative ring theory, to [9] for fundamentals ontopology and to [21] for details on multiplicative theory of ideals.
In this section we give the definition and an alternative characterization of Zariski topology on theprime spectrum Spec(M) of an le-module M . For any submodule element n of M , we consider twodifferent types of varieties V ( n ) and V ∗ ( n ) defined by V ( n ) = { p ∈ Spec ( M ) : n p } ; andV ∗ ( n ) = { p ∈ Spec ( M ) : ( n : e ) ⊆ ( p : e ) } . Then V ( n ) ⊆ V ∗ ( n ) for every submodule element n of M . Proposition 3.1.
Let R M be an le-module. Then(i) V ∗ (0 M ) = X M = V (0 M ) ; ii) V ∗ ( e ) = ∅ = V ( e ) ;(iii) For an arbitrary family of submodule elements { n i } i ∈ I of M ,(a) ∩ i ∈ I V ∗ ( n i ) = V ∗ ( P i ∈ I ( n i : e ) e ) ;(b) ∩ i ∈ I V ( n i ) = V ( P i ∈ I n i ) ;(iv) For any two submodule elements n and l of M ,(a) V ∗ ( n ) ∪ V ∗ ( l ) = V ∗ ( n ∧ l ) ;(b) V ( n ) ∪ V ( l ) ⊆ V ( n ∧ l ) .Proof. ( i ) and ( ii ) are obvious. Also the proofs of ( iii )( b ) and ( iv )( b ) are similar to ( iii )( a ) and( iv )( a ) respectively. Hence we prove here only ( iii )( a ) and ( iv )( a ).( iii )( a ) Let p ∈ ∩ i ∈ I V ∗ ( n i ). Then ( n i : e ) ⊆ ( p : e ) implies that ( n i : e ) e ( p : e ) e p , for all i ∈ I .Consequently P i ∈ I ( n i : e ) e p and so ( P i ∈ I ( n i : e ) e : e ) ⊆ ( p : e ). Hence p ∈ V ∗ ( P i ∈ I ( n i : e ) e )and it follows that ∩ i ∈ I V ∗ ( n i ) ⊆ V ∗ ( P i ∈ I ( n i : e ) e ). Next let p ∈ V ∗ ( P i ∈ I ( n i : e ) e ). Then for any j ∈ I , ( n j : e ) ⊆ (( n j : e ) e : e ) ⊆ ( P i ∈ I ( n i : e ) e : e ) ⊆ ( p : e ) implies that p ∈ V ∗ ( n j ) and so V ∗ ( P i ∈ I ( n i : e ) e ) ⊆ ∩ i ∈ I V ∗ ( n i ). Thus ∩ i ∈ I V ∗ ( n i ) = V ∗ ( P i ∈ I ( n i : e ) e ).( iv )( a ) Now n ∧ l n and n ∧ l l implies that V ∗ ( n ) ∪ V ∗ ( l ) ⊆ V ∗ ( n ∧ l ). Let p ∈ V ∗ ( n ∧ l ).Then ( n ∧ l : e ) ⊆ ( p : e ) implies that ( n : e ) ∩ ( l : e ) ⊆ ( p : e ). Since ( p : e ) is a primeideal, either ( n : e ) ⊆ ( p : e ) or ( l : e ) ⊆ ( p : e ). Hence p ∈ V ∗ ( n ) ∪ V ∗ ( l ) and it follows that V ∗ ( n ∧ l ) ⊆ V ∗ ( n ) ∪ V ∗ ( l ). Therefore V ∗ ( n ) ∪ V ∗ ( l ) = V ∗ ( n ∧ l ).Thus we see that the collection { V ( n ) | n is a submodule element of M } is not closed underfinite unions and hence fails to be the set of all closed subsets of some topology on X M . For any ideal I , Ie is a submodule element of M . Now we see that the subcollection { V ( Ie ) | I is an ideal of R } is closed under finite unions. Lemma 3.2.
Let R M be an le-module. Then for any ideals I and J in R ,(i) V ( Ie ) ∪ V ( J e ) = V (( I ∩ J ) e ) = V (( IJ ) e ) ;(ii) V ∗ ( Ie ) ∪ V ∗ ( J e ) = V ∗ (( I ∩ J ) e ) = V ∗ (( IJ ) e ) . In particular, V ∗ ( re ) ∪ V ∗ ( se ) = V ∗ (( rs ) e ) forany r, s ∈ R .Proof. (i) First I ∩ J ⊆ I implies that ( I ∩ J ) e Ie and so V ( Ie ) ⊆ V (( I ∩ J ) e ). Similarly V ( J e ) ⊆ V (( I ∩ J ) e ), and we have V ( Ie ) ∪ V ( J e ) ⊆ V (( I ∩ J ) e ). Also IJ ⊆ I ∩ J implies that V (( I ∩ J ) e ) ⊆ V (( IJ ) e ). Now let p ∈ V (( IJ ) e ). Then ( IJ ) e p implies that IJ ⊆ ( p : e ). Since6 p : e ) is a prime ideal, either I ⊆ ( p : e ) or J ⊆ ( p : e ). Then either Ie ( p : e ) e p or J e ( p : e ) e p . Hence p ∈ V ( Ie ) ∪ V ( J e ) and it follows that V (( IJ ) e ) ⊆ V ( Ie ) ∪ V ( J e ). Thus V ( Ie ) ∪ V ( J e ) ⊆ V (( I ∩ J ) e ) ⊆ V (( IJ ) e ) ⊆ V ( Ie ) ∪ V ( J e ). This completes the proof.(ii) Similar.We denote, V ( M ) = { V ( n ) : n is a submodule element of M } , V ∗ ( M ) = { V ∗ ( n ) : n is a submodule element of M } , V ′ ( M ) = { V ( Ie ) : I is an ideal of R } . From Proposition 3.1, it follows that there exists a topology, τ ( M ) say, on Spec( M ) having V ( M ) asthe collection of all closed sets if and only if V ( M ) is closed under finite unions. In this case, we callthe topology τ ( M ) the quasi-Zariski topology on Spec( M ). Also from Proposition 3.1, it is evidentthat for any le-module R M there always exists a topology, τ ∗ ( M ) say, on Spec( M ) having V ∗ ( M ) asthe family of all closed sets. This topology τ ∗ ( M ) is called the Zariski topology on Spec( M ). In thisarticle we focus on the basic properties of the Zariski topology τ ∗ ( M ). By Lemma 3.2, it follows that V ′ ( M ) induces a topology, τ ′ ( M ) say, on Spec( M ) for every le-module R M .Now we study the interrelations among these three topologies τ ( M ), τ ∗ ( M ) and τ ′ ( M ). Proposition 3.3.
Let R M be an le-module and n , l are submodule elements of M . If ( n : e ) = ( l : e ) then V ∗ ( n ) = V ∗ ( l ) . The converse is also true if both n and l are prime.Proof. Let p ∈ V ∗ ( n ). Then ( n : e ) ⊆ ( p : e ), i.e, ( l : e ) ⊆ ( p : e ) and hence p ∈ V ∗ ( l ). Thus V ∗ ( n ) ⊆ V ∗ ( l ). Similarly V ∗ ( l ) ⊆ V ∗ ( n ). Therefore V ∗ ( n ) = V ∗ ( l ). Conversely suppose that V ∗ ( n ) = V ∗ ( l ) and both n and l are prime. Let r ∈ ( n : e ). Then re n implies that V ∗ ( n ) ⊆ V ∗ ( re ),i.e, V ∗ ( l ) ⊆ V ∗ ( re ). Since l is a prime submodule element, l ∈ V ∗ ( l ), and so l ∈ V ∗ ( re ). Thus r ∈ ( re : e ) ⊆ ( l : e ). Therefore ( n : e ) ⊆ ( l : e ). Similarly ( l : e ) ⊆ ( n : e ), and hence( n : e ) = ( l : e ). Proposition 3.4.
Let R M be an le-module, n be a submodule element of M and I be an ideal of R .Then(i) V ∗ ( n ) = ∪ P ∈ V R (( n : e )) Spec P ( M ) ;(ii) V ∗ ( n ) = V ∗ (( n : e ) e ) = V (( n : e ) e ) ;(iii) V ( Ie ) = V ∗ ( Ie ) . In particular V ( re ) = V ∗ ( re ) for every r ∈ R . roof. (i) Let p ∈ V ∗ ( n ). Then ( n : e ) ⊆ ( p : e ) and so p ∈ ∪ P ∈ V R (( n : e )) Spec P ( M ), since ( p : e ) itselfa prime ideal. Thus V ∗ ( n ) ⊆ ∪ P ∈ V R (( n : e )) Spec P ( M ). Also let p ∈ ∪ P ∈ V R (( n : e )) Spec P ( M ). Then thereexists a prime ideal P ∈ V R (( n : e )) such that p ∈ Spec P ( M ). This implies that ( n : e ) ⊆ P = ( p : e ), i.e, p ∈ V ∗ ( n ). Hence ∪ P ∈ V R (( n : e )) Spec P ( M ) ⊆ V ∗ ( n ). Therefore V ∗ ( n ) = ∪ P ∈ V R (( n : e )) Spec P ( M ).(ii) Since ( n : e ) e n , V ∗ ( n ) ⊆ V ∗ (( n : e ) e ). Let p ∈ V ∗ (( n : e ) e ). Then (( n : e ) e : e ) ⊆ ( p : e ). Now( n : e ) ⊆ (( n : e ) e : e ) implies that ( n : e ) ⊆ ( p : e ) and so p ∈ V ∗ ( n ). Thus V ∗ (( n : e ) e ) ⊆ V ∗ ( n )and hence V ∗ ( n ) = V ∗ (( n : e ) e ). Let p ∈ V ∗ ( n ). Then ( n : e ) ⊆ ( p : e ) which implies that( n : e ) e ( p : e ) e p , i.e, p ∈ V (( n : e ) e ). Thus V ∗ ( n ) ⊆ V (( n : e ) e ). Also let p ∈ V (( n : e ) e ).Then ( n : e ) e p implies (( n : e ) e : e ) ⊆ ( p : e ). Thus p ∈ V ∗ (( n : e ) e ) = V ∗ ( n ) and hence V (( n : e ) e ) ⊆ V ∗ ( n ). Therefore V ∗ ( n ) = V (( n : e ) e ) = V ∗ (( n : e ) e ).(iii) The proof is omitted since it is easy to prove. Theorem 3.5.
For any le-module R M , the Zariski topology τ ∗ ( M ) on Spec ( M ) is identical with τ ′ ( M ) .Proof. It is suffices to prove that V ∗ ( M ) = V ′ ( M ). Let V ∗ ( n ) be a closed set in V ∗ ( M ) for somesubmodule element n of M . Then by Proposition 3.4, V ∗ ( n ) = V (( n : e ) e ) = V ( Ie ), where ( n : e ) = I ,an ideal of R . Thus every closed set in V ∗ ( M ) is a closed set in V ′ ( M ) and hence V ∗ ( M ) ⊆ V ′ ( M ).Now let V ( Ie ) be a closed set in V ′ ( M ) for some ideal I of R . Again by Proposition 3.4, V ( Ie ) = V ∗ ( Ie ). Since Ie is a submodule element of M , V ( Ie ) is a closed set in V ∗ ( M ). Thus V ′ ( M ) ⊆ V ∗ ( M ).Therefore V ∗ ( M ) = V ′ ( M ).An le-module M is called a top le-module if V ( M ) is closed under finite unions. Hence if M isa top le-module, then τ ( M ) = { X M \ V ( n ) | n is a submodule element of M } becomes a topologyon X M . Theorem 3.6.
For any top le-module R M , the quasi-Zariski topology τ ( M ) on Spec ( M ) is finer thanthe Zariski topology τ ∗ ( M ) = τ ′ ( M ) .Proof. Clearly V ′ ( M ) ⊆ V ( M ). Then by Theorem 3.5, V ∗ ( M ) = V ′ ( M ) ⊆ V ( M ) and hence thequasi-Zariski topology τ ( M ) on Spec( M ) is finer than the Zariski topology τ ∗ ( M ). Let R M be an le-module. Then Ann( M ) is an ideal of R , which allows us to consider the quotient ring R = R/Ann ( M ). The image of every element r ∈ R and every ideal I of R such that Ann ( M ) ⊆ I φ : R → R/Ann ( M ) will be denoted by r and I , respectively.Then for every prime ideal P of R and Ann ( M ) ⊆ P , the ideal P is prime in R . Hence the mapping ψ : X M → X R defined by ψ ( p ) = ( p : e ) for every p ∈ X M is well defined. We call ψ the natural map on X M .In this section we study relationship of X M and X R under the natural map. Here we areinterested in conditions under which ψ is injective, surjective, open, closed, and homeomorphic. Proposition 4.1.
For any le-module M , the natural map ψ of X M is continuous for the Zariskitopologies; more precisely , ψ − ( V R ( I )) = V ( Ie ) for every ideal I of R containing Ann ( M ) .Proof. Let I be an ideal of R containing Ann( M ) and let p ∈ ψ − ( V R ( I ). Then there exists some J ∈ V R ( I ) such that ψ ( p ) = J , i.e, ( p : e ) /Ann ( M ) = J/Ann ( M ). This implies that ( p : e ) = J ⊇ I and so Ie ( p : e ) e p . Hence p ∈ V ( Ie ). Therefore ψ − ( V R ( I )) ⊆ V ( Ie ). Now let q ∈ V ( Ie ).Then I ⊆ ( Ie : e ) ⊆ ( q : e ) implies that I = I/Ann ( M ) ⊆ ( q : e ) /Ann ( M ) = ( q : e ). Hence q ∈ ψ − ( V R ( I )). Thus V ( Ie ) ⊆ ψ − ( V R ( I )). Therefore ψ − ( V R ( I )) = V ( Ie ). Proposition 4.2.
The following conditions are equivalent for any le-module R M :(i) The natural map ψ : X M → X R is injective;(ii) For every p, q ∈ X M , V ∗ ( p ) = V ∗ ( q ) implies that p = q ;(iii) | Spec P ( M ) | for every p ∈ Spec ( R ) .Proof. (i) ⇒ (ii): Let V ∗ ( p ) = V ∗ ( q ). Then ( p : e ) = ( q : e ) which implies that ( p : e ) /Ann ( M ) = ( q : e ) /Ann ( M ). Thus ψ ( p ) = ψ ( q ) and hence p = q , since ψ is injective.(ii) ⇒ (iii): Let p, q ∈ Spec P ( M ), where P ∈ Spec ( R ). Then ( p : e ) = P = ( q : e ) which implies that V ∗ ( p ) = V ∗ ( q ). Hence p = q , by (ii).(iii) ⇒ (i): Let p, q ∈ X M be such that ψ ( p ) = ψ ( q ). Then ( p : e ) /Ann ( M ) = ( q : e ) /Ann ( M ). Thisimplies that ( p : e ) = ( q : e ) = P , say. Thus p, q ∈ Spec P ( M ) and so p = q , by (iii). Therefore ψ isinjective. Theorem 4.3.
Let R M be an le-module and ψ : X M → X R be the natural map of X M . If ψ issurjective, then ψ is both closed and open. More precisely, for every submodule element n of M , ψ ( V ∗ ( n )) = V R ( n : e ) and ψ ( X M − V ∗ ( n )) = X R − V R ( n : e ) . roof. By the Theorem 4.1, we have ψ is a continuous map and ψ − ( V R ( I )) = V ( Ie ), for everyideal I of R containing Ann ( M ). Thus for every submodule element n of M , ψ − ( V R ( n : e )) = V (( n : e ) e ) = V ∗ ( n ). This implies that ψ ( V ∗ ( n )) = ψoψ − ( V R ( n : e )) = V R ( n : e ), since ψ issurjective. Similarly ψ ( X M − V ∗ ( n )) = ψ ( ψ − ( X R ) − ψ − ( V R ( n : e ))) = ψ ( ψ − ( X R − V R ( n : e ))) = ψoψ − ( X R − V R ( n : e )) = X R − V R ( n : e ). Thus ψ is both closed and open. Corollary 4.4.
Let R M be an le-module and ψ : X M → X R be the natural map of X M . Then ψ isbijective if and only if ψ is homeomorphic. A commutative ring R with 1 is said to be a quasi-local ring if it has a unique maximal ideal.
Theorem 4.5.
Let R M be an le-module and ψ : X M → X R be the surjective natural map of X M .Then the following statements are equivalent:(i) X M = Spec ( M ) is connected;(ii) X R = Spec ( R ) is connected;(iii) The ring R contains no idempotent other than and .Consequently, if either R is a quasi-local ring or Ann ( M ) is a prime ideal of R , then both X M and X R are connected.Proof. (i) ⇒ (ii): From Theorem 4.1, we have that ψ is a continuous map. Then (ii) follows fromthe fact that ψ is surjective and continuous image of a connected space is connected.(ii) ⇒ (i): Let X R be connected. If possible assume that X M is disconnected. Then X M mustcontain a non-empty proper subset Y which is both open and closed. By Theorem 4.3, ψ ( Y ) is anon-empty subset of X R that is both open and closed. We assert that ψ ( Y ) is a proper subset of X R . Since Y is open, Y = X M − V ∗ ( n ) for some submodule element n of M . Then by Theorem4.3, ψ ( Y ) = ψ ( X M − V ∗ ( n )) = X R − V R ( n : e ). Therefore, if ψ ( Y ) = X R , then V R ( n : e ) = ∅ .Now suppose that ( n : e ) = R . Then ( n : e ) is a proper ideal of R and so contained in a maximalideal, say P of R , which is also a prime ideal of R . Thus ( n : e ) ⊆ P and hence P ∈ V R ( n : e ), i.e, V R ( n : e ) = ∅ , a contradiction. Thus ( n : e ) = R , i.e, n = e . This implies that Y = X M − V ∗ ( n ) = X M − V ∗ ( e ) = X M , which is an absurd since Y is a proper subset of X M . Thus ψ ( Y ) is a propersubset of X R and hence X R is disconnected, a contradiction. Therefore X M = Spec ( M ) is connected.The equivalence of (ii) and (iii) is well-known [8].10 A base for the Zariski topology on Spec(M)
For any element r of a ring R , the set D r = X R − V R ( rR ) is open in X R and the family { D r : r ∈ R } forms a base for the Zariski topology on X R . Each D r , in particular D = X R is known to bequasi-compact. In [22], Chin-Pi Lu, introduced a base for the Zariski topology on Spec( M ) for any R -module M , which is similar to that on X R . In this section, we introduce a base for the Zariskitopology on X M for any le-module R M .For each r ∈ R we define, X r = X M − V ∗ ( re ).Then every X r is an open set in X M . Note that X = ∅ and X = X M . Proposition 5.1.
Let R M be an le-module with the natural map ψ : X M → X R and r ∈ R . Then(i) ψ − ( D r ) = X r ; and(ii) ψ ( X r ) ⊆ D r ; the equality holds if ψ is surjective.Proof. (i): ψ − ( D r ) = ψ − ( X R − V R ( rR )) = ψ − ( X R ) − ψ − ( V R ( rR )) = ψ − ( V R (0)) − ψ − ( V R ( rR )) = V (0 M ) − V ( rRe ) = X M − V ( re ) = X M − V ∗ ( re ) = X r , by Proposition 4.1 and Proposition 3.4(iii).(ii) follows from (i).Now we have a useful lemma which will be used in the next theorem: Lemma 5.2.
Let R M be an le-module.(i) For every r, s ∈ R , X rs = X r ∩ X s .(ii) For any ideal I in R , V ∗ ( Ie ) = ∩ a ∈ I V ∗ ( ae ) .Proof. (i) X rs = X M − V ∗ (( rs ) e ) = X M − ( V ∗ ( re ) ∪ V ∗ ( se )) = ( X M − V ∗ ( re )) ∩ ( X M − V ∗ ( se )) = X r ∩ X s , by Lemma 3.2(ii).(ii) Let p ∈ V ∗ ( Ie ). Then ( Ie : e ) ⊆ ( p : e ). Now for all a ∈ I , ae Ie implies that ( ae : e ) ⊆ ( Ie : e ) ⊆ ( p : e ). Thus p ∈ V ∗ ( ae ) for all a ∈ I and so p ∈ ∩ a ∈ I V ∗ ( ae ). Hence V ∗ ( Ie ) ⊆ ∩ a ∈ I V ∗ ( ae ).Also let p ∈ ∩ a ∈ I V ∗ ( ae ). Then for all a ∈ I , p ∈ V ∗ ( ae ) = V ( ae ), by Proposition 3.4, which impliesthat ae p . Thus for any k ∈ N and a , a , · · · , a k ∈ I , a e + a e + · · · + a k e p and hence Ie p . Then ( Ie : e ) ⊆ ( p : e ) and so p ∈ V ∗ ( Ie ). Hence ∩ a ∈ I V ∗ ( ae ) ⊆ V ∗ ( Ie ). Therefore V ∗ ( Ie ) = ∩ a ∈ I V ∗ ( ae ). Theorem 5.3.
Let R M be an le-module. Then the set B = { X r : r ∈ R } forms a base for the Zariskitopology on X M which may be empty. roof. If X M = ∅ , then B = ∅ and theorem is trivially true in this case. Let X M = ∅ and U bean any open set in X M . Then U = X M − V ∗ ( Ie ) for some ideal I of R since V ∗ ( M ) = V ′ ( M ) = { V ∗ ( Ie ) = V ( Ie ) : I is an ideal of R } , by Proposition 3.4. By above lemma V ∗ ( Ie ) = ∩ a ∈ I V ∗ ( ae ).Hence U = X M − V ∗ ( Ie ) = X M − ∩ a ∈ I V ∗ ( ae ) = ∪ a ∈ I ( X M − V ∗ ( ae )) = ∪ a ∈ I X a . Thus B is a basefor the Zariski topology on X M .A topological space T is called quasi-compact if every open cover of T has a finite subcover.Every finite space is quasi-compact, and more generally every space in which there is only a finitenumber of open sets is quasi-compact. A subset Y of a topological space T is said to be quasi-compactif the subspace Y is quasi-compact. By a quasi-compact open subset of T we mean an open subset of T which is quasi-compact. To avoid ambiguity, we would like to mention that a compact topologicalspace is a quasi-compact Hausdorff space. Quasi-compact spaces are of use mainly in applicationsof topology to algebraic geometry and are seldom featured in other mathematical theories, where onthe contrary compact spaces play an important role in different branches of mathematics. To keepuniformity in terminology we continue with the term quasi-compact. Theorem 5.4.
Let R M be an le-module and the natural map ψ : X M → X R is surjective. Then thefollowing statements hold:(i) The open set X r in X M for each r ∈ R is quasi-compact. In particular, the space X M is quasi-compact.(ii) The quasi-compact open sets of X M are closed under finite intersection and form an open base.Proof. (i) Since B = { X r : r ∈ R } forms a base for the zariski topology on X M by Theorem 5.3, forany open cover of X r , there is a family { r λ : λ ∈ Λ } of elements of R such that X r ⊆ ∪ λ ∈ Λ X r λ . ByProposition 5.1.( ii ), D r = ψ ( X r ) ⊆ ∪ λ ∈ Λ ψ ( X r λ ) = ∪ λ ∈ Λ D r λ . Since D r is quasi-compact, there existsa finite subset Λ ′ of Λ such that D r ⊆ ∪ λ ∈ Λ ′ D r λ . Hence X r = ψ − ( D r ) ⊆ ∪ λ ∈ Λ ′ X r λ , by Proposition5.1.( i ). Thus for each r ∈ R , X r is quasi-compact.(ii) To prove the theorem it suffices to prove that the intersection of two quasi-compact open setsof X M is a quasi-compact set. Let C = C ∩ C , where C , C are quasi-compact open sets of X M .Since B = { X r : r ∈ R } is an open base for the Zariski topology on X M , each C i , i = 1 ,
2, is afinite union of members of B . Then by Proposition 5.2, it follows that C is also a finite union ofmembers of B . Let C = ∪ ni =1 X r i and let Ω be any open cover of C . Then Ω also covers each X r i which is quasi-compact by (i). Hence each X r i , has a finite subcover of Ω and so does C . Thus C isquasi-compact. The other part of the theorem follows from the existence of the open base B .12 Irreducible closed subsets and generic points
For each subset Y of X M , we denote the closure of Y in Zariski topology on X M by Y , and meet ofall elements of Y by ℑ ( Y ), i.e. ℑ ( Y ) = ∧ p ∈ Y p . One can check that ℑ ( Y ) is a submodule element of M . For each subset Y of Spec ( R ), we denote the intersection ∩ P ∈ Y P of all elements of Y by ℑ R ( Y ) Proposition 6.1.
Let R M be an le-module and Y ⊆ X M . Then V ∗ ( ℑ ( Y )) = Y . Hence Y is closedif and only if V ∗ ( ℑ ( Y )) = Y .Proof. To prove the result it is sufficient to prove that V ∗ ( ℑ ( Y )) is the smallest closed subset of X M containing Y . Now for all p ∈ Y , ℑ ( Y ) = ∧ p ∈ Y p p implies that ( ℑ ( Y ) : e ) ⊆ ( p : e ),i.e., p ∈ V ∗ ( ℑ ( Y )). Hence Y ⊆ V ∗ ( ℑ ( Y )). Now let V ∗ ( n ) be any closed subset of X M such that Y ⊆ V ∗ ( n ). Then for every p ∈ Y , ( n : e ) ⊆ ( p : e ) and so ( n : e ) ⊆ ∩ p ∈ Y ( p : e ) = ( ∧ p ∈ Y p : e ) =( ℑ ( Y ) : e ). Also let q ∈ V ∗ ( ℑ ( Y )). Then ( n : e ) ⊆ ( ℑ ( Y ) : e ) ⊆ ( q : e ) implies that q ∈ V ∗ ( n ). Thus V ∗ ( ℑ ( Y )) ⊆ V ∗ ( n ). Therefore V ∗ ( ℑ ( Y )) = Y .For an le-module R M , we denote Φ = { ( p : e ) | p ∈ X M } . Then Φ ⊆ X R , by Lemma 2.2. We say P is a maximal element of Φ if for any Q ∈ Φ, P ⊆ Q implies that P = Q . Recall that a topologicalspace is a T -space if and only if every singleton subset is closed. Proposition 6.2.
Let R M be an le-module and p ∈ X M . Then(i) { p } = V ∗ ( p ) ;(ii) For any q ∈ X M , q ∈ { p } if and only if ( p : e ) ⊆ ( q : e ) if and only if V ∗ ( q ) ⊆ V ∗ ( p ) ;(iii) The set { p } is closed in X M if and only if ( a ) P = ( p : e ) is a maximal element of Φ , and ( b ) Spec P ( M ) = { p } , i.e, | Spec P ( M ) | = 1 ;(iv) Spec ( M ) is a T -space if and only if(a) P = ( p : e ) is a maximal element of Φ for every p ∈ X M , and(b) | Spec P ( M ) | for every P ∈ Spec ( R ) .Proof. (i) It follows from Proposition 6.1 by taking Y = { p } .(ii) It is an obvious result of (i).(iii) Let { p } be closed in X M . Then by (i), { p } = V ∗ ( p ). To show P = ( p : e ) is a maximal element13f Φ, let Q ∈ Φ be such that P ⊆ Q . Since Q ∈ Φ there is a prime submodule element q of X M suchthat ( q : e ) = Q . Then ( p : e ) ⊆ ( q : e ) which implies that q ∈ V ∗ ( p ) = { p } . Thus p = q and so P = Q . For (ii) suppose that q be any element of Spec P ( M ). Then ( q : e ) = P = ( p : e ) implies that q ∈ V ∗ ( p ) = { p } and hence q = p .Conversely, we assume that the conditions (a) and (b) hold. Let q ∈ V ∗ ( p ). Then ( p : e ) ⊆ ( q : e )and so P = ( p : e ) = ( q : e ), by (a). Now (b) implies that p = q , i.e, V ∗ ( p ) ⊆ { p } . Also { p } ⊆ V ∗ ( p ).Thus { p } = V ∗ ( p ), i.e, { p } is closed in X M by (i).(iv) Note that (b) is equivalent to that | Spec P ( M ) | = 1 for every P ∈ Φ. Thus, by (iii), it followsthat { p } is closed in X M for every p ∈ X M . Hence X M is a T -space.A topological space T is called irreducible if for every pair of closed subsets T , T of T , T = T ∪ T implies T = T or T = T . A subset Y of T is irreducible if it is irreducible as a subspaceof T . By an irreducible component of a topological space T we mean a maximal irreducible subsetof T . Also if a subset Y of a topological space T is irreducible, then its closure Y is so. Since everysingleton subset of X M is irreducible, its closure is also irreducible. Now by Proposition 6.2, we havethe following result: Corollary 6.3. V ∗ ( p ) is an irreducible closed subset of X M for every prime submodule element p ofan le-module R M . It is well known that a subset Y of Spec( R ) for any ring R is irreducible if and only if ℑ R ( Y )is a prime ideal of R [8]. Let M be a left R -module. Then a subset Y of Spec( M ) is irreducible if ℑ M ( Y ) = ∩ P ∈ Y P is a prime submodule of M , but the converse is not true in general [22]. In thefollowing result we show that the situation in an le-module R M is similar to the modules over a ring.Interestingly, the converse of this result in an le-module R M is directly associated with the ring R . Proposition 6.4.
Let R M be an le-module and Y ⊆ X M . If ℑ ( Y ) is a prime submodule element of M then Y is irreducible. Conversely, if Y is irreducible then Ψ = { ( p : e ) | p ∈ Y } is an irreduciblesubset of Spec ( R ) , i.e, ℑ R (Ψ) = ( ℑ ( Y ) : e ) is a prime ideal of R .Proof. Let ℑ ( Y ) be a prime submodule element of M . Then by Corollary 6.3, V ∗ ( ℑ ( Y )) = Y is irreducible and hence Y is irreducible. Conversely, suppose that Y is irreducible. Since ψ iscontinuous by Proposition 4.1, the image ψ ( Y ) of Y under the natural map ψ of X M is an irreduciblesubset of X R . Hence ℑ R ( ψ ( Y )) = ( ℑ ( Y ) : e ) is a prime ideal of X R . Thus ℑ R (Ψ) = ( ℑ ( Y ) : e ) is aprime ideal of R so that Ψ is an irreducible subset of Spec( R ).14lso we have some other characterization of the irreducible subsets of X M . Proposition 6.5.
Let R M be an le-module. Then the following statements hold:(i) If Y = { p i | i ∈ I } is a family of prime submodule elements of M which is totally ordered by“ ”, then Y is irreducible in X M .(ii) If Spec P ( M ) = ∅ for some P ∈ Spec ( R ) , then(a) Spec P ( M ) is irreducible, and(b) Spec P ( M ) is an irreducible closed subset of X M if P is a maximal ideal of R .(iii) Let Y ⊆ X M be such that ( ℑ ( Y ) : e ) = P is a prime ideal of R . Then Y is irreducible ifSpec P ( M ) = ∅ .Proof. (i) It is suffices to show that ℑ ( Y ) = ∧ i ∈ I p i is a prime submodule element. Let r ∈ R and n ∈ M be such that n (cid:2) ℑ ( Y ) and r / ∈ ( ℑ ( Y ) : e ) = ( ∧ i ∈ I p i : e ) = ∩ i ∈ I ( p i : e ). Then there exists l and k such that n (cid:2) p l and r / ∈ ( p k : e ). Since Y is totally ordered, either p l p k or p k p l . Let p l p k . Then r / ∈ ( p l : e ) and n (cid:2) p l implies that rn (cid:2) p l since p l is a prime submodule element.Thus rn (cid:2) ∧ i ∈ I p i = ℑ ( Y ). Hence ℑ ( Y ) is a prime submodule element.(ii) Let Spec P ( M ) = ∅ for some P ∈ Spec ( R ). Then ℑ ( Spec P ( M )) is a proper submodule elementof M .(a) Assume that r ∈ R and n ∈ M be such that rn ℑ ( Spec P ( M )) and n (cid:10) ℑ ( Spec P ( M )).Since n (cid:10) ℑ ( Spec P ( M )), there exists p ∈ X M with ( p : e ) = P such that n (cid:10) p . Now rn ℑ ( Spec P ( M )) p and n (cid:10) p implies that re p , since p is a prime submodule element. Thus r ∈ ( p : e ) = P = ( ℑ ( Spec P ( M )) : e ). Therefore ℑ ( Spec P ( M )) is a prime submodule element andhence Spec P ( M ) is irreducible.(b) We prove that Spec P ( M ) = V ∗ ( P e ) so that Spec P ( M ) is closed. Let q ∈ Spec P ( M ). then( q : e ) = P ⊆ ( P e : e ). Since P is a maximal ideal of R , ( P e : e ) = P = ( q : e ), and so q ∈ V ∗ ( P e ).Thus
Spec P ( M ) ⊆ V ∗ ( P e ). Also let q ∈ V ∗ ( P e ). Then P ⊆ ( P e : e ) ⊆ ( q : e ) which implies that( q : e ) = P , since P is a maximal ideal. Thus q ∈ Spec P ( M ) and hence V ∗ ( P e ) ⊆ Spec P ( M ).Therefore Spec P ( M ) = V ∗ ( P e ).(iii) Let p be any element of Spec P ( M ). Then ( p : e ) = P = ( ℑ ( Y ) : e ) which implies that V ∗ ( p ) = V ∗ ( ℑ ( Y )) = Y , by Proposition 6.1. Thus Y is irreducible and hence Y is irreducible.Let Y be a closed subset of a topological space T . An element y ∈ Y is called a generic point of Y if Y = { y } . Proposition 6.2 shows that every prime submodule element p of M is a generic point15f the irreducible closed subset V ∗ ( p ) in X M . Now we prove that every irreducible closed subset of X M has a generic point. Theorem 6.6.
Let R M be an le-module and the natural map ψ : X M → X R be surjective. Then thefollowing statements hold.(i) Then Y ⊆ X M is an irreducible closed subset of X M if and only if Y = V ∗ ( p ) for some p ∈ X M .Every irreducible closed subset of X M has a generic point.(ii) The correspondence p V ∗ ( p ) is a surjection of X M onto the set of irreducible closed subsetsof X M .(iii) The correspondence V ∗ ( p ) ( p : e ) is a bijection of the set of irreducible components of X M onto the set of minimal prime ideals of R .Proof. (i) Let Y be an irreducible closed subset of X M . Then Y = V ∗ ( n ) for some submodule element n of M . By Proposition 6.4, we have ( ℑ ( V ∗ ( n )) : e ) = ( ℑ ( Y ) : e ) = P is a prime ideal of R . Then P/Ann ( M ) ∈ X R . Since ψ is surjective, there exists a prime submodule element p of X M such that ψ ( p ) = P/Ann ( M ), i.e, ( p : e ) /Ann ( M ) = P/Ann ( M ) and hence ( p : e ) = P = ( ℑ ( V ∗ ( n )) : e ).Therefore V ∗ ( ℑ ( V ∗ ( n ))) = V ∗ ( p ). Since V ∗ ( n ) is closed, V ∗ ( ℑ ( V ∗ ( n ))) = V ∗ ( p ) implies V ∗ ( n ) = V ∗ ( p ), by Proposition 6.1. Thus Y = V ∗ ( n ) = V ∗ ( p ).Converse part follows from Corollary 6.3.Now it follows from V ∗ ( p ) = { p } that every irreducible closed subset of X M has a generic point.(ii) Follows from (i).(iii) First assume that V ∗ ( p ) is an irreducible component of X M . Then V ∗ ( p ) is maximal in the set { V ∗ ( q ) : q ∈ X M } . Clearly ( p : e ) is a prime ideal of R . To show ( p : e ) is minimal let P be a primeideal of R such that P ⊆ ( p : e ). Then P ⊆ ( p : e ). Since ψ is surjective there exists q ∈ X M suchthat ψ ( q ) = P , i.e, ( q : e ) = P . Therefore ( q : e ) = P ⊆ ( p : e ) and hence V ∗ ( p ) ⊆ V ∗ ( q ). Since V ∗ ( p ) is maximal in the set { V ∗ ( q ) : q ∈ X M } , V ∗ ( p ) = V ∗ ( q ). Thus ( p : e ) = ( q : e ) = P and so P = ( p : e ).Next let P be a minimal prime ideal of R . Since ψ is surjective there exists p ∈ X M such that( p : e ) = P . From Corollary 6.3, V ∗ ( p ) is an irreducible subset of X M . Now let q ∈ X M be suchthat V ∗ ( p ) ⊆ V ∗ ( q ). Then ( q : e ) ⊆ ( p : e ) = P which implies that ( q : e ) ⊆ P . Since P is minimal,( q : e ) = P . Thus ( q : e ) = P = ( p : e ) and so V ∗ ( p ) = V ∗ ( q ). Therefore V ∗ ( p ) is an irreduciblecomponent of X M . This completes the proof. 16 X M as a spectral space A topological space T is called spectral if it is T , quasi-compact, the quasi-compact open subsetsof T are closed under finite intersection and form an open basis, and each irreducible closed subsetof T has a generic point. Importance of spectral topological space is that a topological space T ishomeomorphic to Spec( R ) for some commutative ring R if and only if T is spectral. Any closedsubset of a spectral topological space with the induced topology is spectral.From Theorem 5.4 and Theorem 6.6, it follows that X M satisfies all the conditions to be aspectral space except being T .Now we prove some equivalent characterizations for X M to be spectral. Theorem 7.1.
Let R M be an le-module and the natural map ψ : X M → X R be surjective. Then thefollowing conditions are equivalent:(i) X M is a spectral space;(ii) X M is a T -space;(iii) For every p, q ∈ X M , V ∗ ( p ) = V ∗ ( q ) implies that p = q ;(iv) | Spec P ( M ) | for every p ∈ Spec ( R ) ;(v) ψ is injective;(vi) X M is homeomorphic to X R .Proof. Equivalence of (iii), (iv) and (v) follows from Proposition 4.2 and equivalence of (v) and (vi)follows from Corollary 4.4.( i ) ⇒ ( iii ): Let p, q ∈ X M be such that V ∗ ( p ) = V ∗ ( q ). Then, by Proposition 6.2, { p } = V ∗ ( p ) = V ∗ ( q ) = { q } . Since X M is spectral, it is T . Hence the closures of distinct points in X M are distinctand it follows that p = q .( iii ) ⇒ ( ii ): Let p, q ∈ X M be two distinct prime submodule elements. Then V ∗ ( p ) = V ∗ ( q ) whichimplies that { p } 6 = { q } , by Proposition 6.2. Hence X M is T .( ii ) ⇒ ( i ): This follows from Theorem 5.4 and Theorem 6.6.An le-module R M is called a multiplication le-module if every submodule element n of M canbe expressed as n = Ie , for some ideal I of R . Let n be a submodule element of a multiplication17e-module R M . Then there exists an ideal I of R such that n = Ie . For each k ∈ N and a i ∈ ( n : e ),we have a e + a e + · · · + a k e n + n + · · · + n ( ktimes ) = n Hence ( n : e ) e n . Also n = Ie implies I ⊆ ( n : e ) which implies that n = Ie ( n : e ) e . Thus( n : e ) e = n . Theorem 7.2.
Let R M be a multiplication le-module and the natural map ψ : X M → X R be surjec-tive. Then X M is a spectral space.Proof. First we show that X M is T . Let l and n be two distinct elements of X M . If possible let V ∗ ( l ) = V ∗ ( n ). Then ( l : e ) = ( n : e ) which implies that l = ( l : e ) e = ( n : e ) e = n , a contradiction.Thus { l } = V ∗ ( l ) = V ∗ ( n ) = { n } , by Proposition 6.2. Hence X M is T . Thus the theorem followsfrom Theorem 5.4 and Theorem 6.6.Thus assuming surjectivity of the natural map ψ : X M −→ X R gives us several equivalentcharacterizations for X M to be spectral. In other words, surjectivity of ψ is a very strong conditionto imply that X M is spectral. Now we prove some conditions for X M to be spectral, which do notassume surjectivity of the natural map ψ : X M −→ X R . Theorem 7.3.
Let R M be an le-module and ψ : X M −→ X R be the natural map of X M such that theimage Im ψ of ψ is a closed subset of X R . Then X M is a spectral space if and only if ψ is injective.Proof. Let Y = Im ψ , Since X R is a spectral space and Y is a closed subset of X R , Y is spectralfor the induced topology. Suppose that ψ is injective. Then ψ : X M → Y is a bijection and byProposition 4.1, ψ : X M → Y is continuous. Now we show that ψ is closed. Let V ∗ ( n ) be a closedsubset of X M for some submodule element n of M . Then Y ′ = Y ∩ V R (( n : e )) is a closed subset of Y . Also ψ − ( Y ′ ) = ψ − ( Y ∩ V R (( n : e ))) = ψ − ( Y ) ∩ ψ − ( V R (( n : e ))) = X M ∩ ψ − ( V R (( n : e ))) = ψ − ( V R (( n : e ))) = V (( n : e ) e ) = V ∗ ( n ) by Proposition 4.1. Since ψ : X M → Y is surjective, ψ ( V ∗ ( n )) = ψ ( ψ − ( Y ′ )) = Y ′ , a closed subset of Y . Thus ψ : X M → Y is a homeomorphism andhence X M is a spectral space. Conversely, let X M be a spectral space. Then X M is a T -space andso ψ is injective by Theorem 7.1. Theorem 7.4.
Let R M be an le-module such that Spec ( M ) = X M is a non-empty finite set. Then X M is a spectral space if and only if | Spec P ( M ) | for every P ∈ Spec ( R ) . roof. The finiteness of | X M | implies that X M is quasi-compact and the quasi-compact open subsetsof X M are closed under finite intersection and forms an open base. We show that every irreducibleclosed subset of X M has a generic point. Let Y = { p , p , · · · , p n } be an irreducible closed subsetof X M . Then Y = Y = V ∗ ( ℑ ( Y )) = V ∗ ( p ∧ p ∧ · · · ∧ p n ) = V ∗ ( p ) ∪ V ∗ ( p ) ∪ · · · ∪ V ∗ ( p n ) = { p } ∪ { p } ∪ · · · ∪ { p n } . Since Y is irreducible, Y = { p i } for some i . Therefore by Hochster‘scharacterization of spectral spaces, X M is a spectral space if and only if X M is a T -space. Henceby Theorem 7.1, it follows that X M is a spectral space if and only if | Spec P ( M ) | P ∈ Spec ( R ). References [1] J. Abuhlail, A dual Zariski topology for modules, Topology Appl. 158 (2011) 457-467.[2] D. D. Anderson, Abstract commutative ideal theory without chain condition, Algebra Universalis6 (1976) 131-145.[3] D. D. Anderson and E. W. Johnson, Abstract ideal theory from Krull to the present, in: Idealtheoretic methods in commutative algebra(Columbia, MO, 1999), Lecture notes in Pure andAppl. Math., Marcel Dekkar, New York, 220 (2001) 27-47.[4] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-WesleyPublishing Co., 1969.[5] M. Behboodi, A generalization of the classical krull dimension for modules, J. Algebra 305 (2006)1128-1148.[6] M. Behboodi and M. R. Haddadi, Classical Zariski topology of modules and spectral spaces I,International Electronic Journal of Algebra 4 (2008) 104-130.[7] A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian latticemodules, Communicated.[8] N. Bourbaki, Commutative Algebra, Springer-Verlag, 1998.[9] N. Bourbaki, General Topology, Part I, Addison-Wesley, 1966.[10] R. P. Dilworth, Non-commutative residuated lattices, Trans. Amer. Math. Soc. 46 (1939) 426-444.[11] R. P. Dilworth, Abstract commutative ideal theory, Pacific J. Math. 12 (1962) 481-498.[12] T. Duraivel, Topology on spectrum of modules, J. Ramanujan Math. Soc. 9 (1) (1994) 25-34.[13] D. Hassanzadeh-Lelekaami and H. Roshan-Shekalgourabi, Prime submodules and a sheaf on theprime spectra of modules, Comm. Algebra 42 (7) (2014) 3063-3077.1914] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 137 (1969)43-60.[15] J. A. Johnson, a-adic completions of Noetherian lattice modules, Fund. Math. 66 (1970) 347-373.[16] J. A. Johnson, Quotients in Noetherian lattice modules, Proceedings of the American Mathe-matical Society 28 (1) (1971) 71-74.[17] J. A. Johnson, Noetherian lattice modules and semi-local comletions, Fundamenta Mathematicae73 (1971) 93-103.[18] E. W. Johnson and J. A. Johnson, Lattice modules over semi-local Noether lattices, Fund. Math.68 (1970) 187-201.[19] E. W. Johnson and J. A. Johnson, Lattice Modules over principal element domains, Comm. inAlgebra 31 (7) (2003) 3505-3518.[20] W. Krull, Axiomatische begr¨ uu