Also, for various types of modules M, we obtain some conditions under whichM axR(M) is homeo- morphic with the maximal ideal space of some ring

(1)

TOPOLOGY

H. ANSARI-TOROGHY AND S. KEYVANI

Abstract. For any module M over a commutative ring R,SpecR(M) (resp.

M axR(M)) ofM is the collection of all prime (resp. maximal) submodules.

In this article we investigate the interplay between the topological properties ofM axR(M) and module theoretic properties ofM. Also, for various types of modules M, we obtain some conditions under whichM axR(M) is homeomorphic with the maximal ideal space of some ring.

1. Introduction

Throughout this article, R is a commutative ring with non zero identity and M is a unitary R-module. For any ideal I of R containing AnnR(M), ¯R and ¯I denoteR/Ann(M) andI/Ann(M) respectively. AlsoN,Z, andQdenote the set of positive integers, the ring of integers, and the field of rational numbers respectively.

Moreover the notation ”⊂” will denote the strict inclusion.

ForM as anR-module andN a submodule, we recall thecolon ideal ofM into N, (N:M) ={r∈R|rM⊆N}.

A submoduleP ofM is said to be aprime submodule ifP 6=M and whenever r ∈ R and e ∈ M satisfy re ∈ P, then r ∈ P or e ∈ (P : M). If P is a prime submodule, then (P : M) is a prime ideal of R. Moreover if Q is a maximal submodule ofM, thenQis a prime submodule and (Q:M) is a maximal ideal of R[9] and [10].

IfSpecR(M)6=∅(resp. M axR(M)6=∅), the mappingψ:SpecR(M)→Spec( ¯R) (resp. φ : M ax_R(M) → M ax( ¯R)) such that ψ(P) = (P:M) (resp. φ(Q) = (Q:M)) for every P ∈ SpecR(M) (resp. Q ∈ M axR(M)), is called the natural map ofSpecR(M) (resp. M axR(M)) [11] (resp. [1]).

M is said to beprimeful (resp. M ax-surjective) if eitherM = (0) or M 6= (0) and the natural map ofSpec_R(M) (resp. M ax_R(M)) is surjective [12] (resp. [1]).

M is said to beX-injective if either Spec_R(M) = ∅ or Spec_R(M)6=∅ and the natural map ofSpec_R(M) is injective [2].

TheZariski topology onX=Spec_R(M) is the topologyτ_M described by taking the set Z(M) ={V_M(N)|N is a submodule ofM} as the set of closed sets ofX, where

VM(N) ={P ∈X|(P :M)⊇(N :M)}.

WhenM =R,τM =τR is the well known Zariski topology onSpec(R) [11].

Date: December 25, 2012.

2000Mathematics Subject Classification. 13E05, 13C99.

Key words and phrases. maximal submodule, M ax-injective module, M ax-spectral space, Zariski topology.

1

(2)

There exists a topology onM axR(M) having Z^m(M) ={V_M^m(N)|N is a submodule ofM}as the set of closed sets ofM axR(M), where

V_M^m(N) ={Q∈M axR(M)|(Q:M)⊇(N :M)}.

We denote this topology by τ_M^m. In fact τ_M^m is the same as the subspace topology induced byτ_M onM ax_R(M). WhenM =R, this topology is denoted byτ_R^m and for every idealI ofR, we have

V_R^m(I) ={q∈M ax(R)|q⊇I}.

In the rest of this articleSpec_R(M) (resp. M ax_R(M)) is always equipped with the Zariski topologyτ_M (resp. τ_M^m) andM ax_R(M) is assumed to be a non-empty subset ofSpec_R(M).

The present authors introduced the concept of M ax-injective modules and in- vestigated some important properties of this family of modules. AnR-modulesM is called M ax-injective if the natural map of M axR(M) is injective [4]. Clearly, everyX-injective module isM ax-injective.

A topological spaceW is said to beM ax-spectral if it is homeomorphic with the maximal ideal space of some ring (see Definition 3.17). M ax-spectral spaces have been characterized by Hochster in [8, p. 57, Proposition 11].

In this article, we investigate the interplay between the topological properties of M axR(M) and module theoretic properties ofM (see Proposition 3.2, Theorem 3.6, Theorem 3.13, Corollary 3.15, Proposition 3.19, and Theorem 3.24). Theorem 3.14 provides useful information about the relationship between topological properties of M ax_R(M) andM ax( ¯R). Also we consider the conditions under whichM ax_R(M) is a Noetherian topological space (see Proposition 3.2, Theorem 3.6, Theorem 3.14, and Corollary 3.15). Moreover, we study the topological spaceM ax_R(M) from the point of view ofM ax-spectral spaces (see Theorem 3.24). It is shown that if M is aM ax-injective module over a PID, thenM axR(M) is aM ax-spectral topological space (see Theorem 3.24 (g)). These results enable us to provide a large family of modules such that their maximal submodules areM ax-spectral.

2. Preliminaries

In this section we review some preliminary results which will be needed in next section.

Definition 2.1. For a topological space X, we recall

(a) X is quasi compact if it satisfies one of the following two equivalent conditions.

(1) Every collection of open subsets whose union is X contains a finite subcollection whose union isX.

(2) Every collection of closed subsets whose intersection is empty set contains a finite subcollection whose intersection is empty set (see [15, Definition 2.135]).

(b) X is said to be Noetherian if the open subsets ofX satisfy the ascending chain condition (or maximal condition). (see [6, Chap. 6, Example 5]).

(c) X is said to be connected if it is not the unionX =X0∪X1of two disjoint closed non-empty subsetsX0 andX1 (see [15, Definition 2.105]).

(d) Xis said to be irreducible ifXis not the union of two proper closed subsets.

ForX⁰⊆X,X⁰is irreducible if it is irreducible as a space with the relative

(3)

topology. This is equivalent to say that, if F, G are closed subsets ofX such thatX⁰⊆F∪G, thenX⁰⊆F or X⁰⊆G(see [7, Ch. II, p. 119]).

(e) A maximal irreducible subset of X is called an irreducible component of X. It is well known that every irreducible component ofX is closed inX (see [7, Ch. II]).

Remark 2.2. LetX andY be two topological spaces.

(a) Let f be a continuous mapping fromX toY.

(1) If X is a connected (resp. quasi compact) topological space, then f(X) is a connected (resp. quasi compact) topological space (see [15, Theorem 2.107 and Theorem 2.138]).

(2) For every irreducible subset E of X, f(E) is an irreducible subset of Y (see [7, Ch. II]).

(b) IfX is a Noetherian topological space, then every subspace ofX is a Noe- therian topological space, andX is a quasi compact topological space (see [6, Chap. 6, Exc. 5]).

(c) Every Noetherian topological space has only finitely many irreducible components (see [7, p. 124, Proposition 10]).

(d) Closed subspaces of quasi compact topological spaces are quasi compact (see [15, Theorem 2.137]).

(e) Every finite topological space is quasi compact (see [15, p. 51]).

(f) Closure of any connected (resp. irreducible) subspace is connected (resp.

irreducible) (see [15, Corollary 2.112] and [7, Ch. II]).

(g) Let A and B be subsets of X such that A ⊆ B ⊆X, where B is closed in X and equipped with the relative topology. Then A is an irreducible closed subset ofB if and only ifAis an irreducible closed subset ofX (see Definition 2.1 (d)).

3. Main results

As it was mentioned before, SpecR(M) (resp. M axR(M)) is always equipped with Zariski topologyτM (resp. τ_M^m).

Lemma 3.1. Let M be an R-module and let φ : M ax_R(M)→ M ax( ¯R) be the natural map ofM ax_R(M). Then the following hold.

(a) φis a continuous map.

(b) IfM isM ax-surjective, thenφis closed and open mapping.

Proof. (a) This follows from the fact that φ⁻¹(V_R^m_¯ ( ¯I)) =V_M^m(IM) for every ideal I ofRcontainingAnn(M).

(b) LetNbe a submodule ofM and letV_M^m(N) be a closed subset ofM axR(M).

Then as in the proof part (a), we have

φ⁻¹(V_R_¯^m((N :M))) =V_M^m((N :M)M) =V_M^m(N).

Henceφ(V_M^m(N)) =V_R_¯^m((N:M)) becauseφis surjective. Alsoφis open by similar arguments and the proof is completed.

A topological space W is a cofinite topological space when its open sets are empty andW and all subsets with a finite complement. This topology is denoted byτ^{f c}.

(4)

Proposition 3.2. Let R be a ring such that the intersection of every infinite collection of maximal ideals of R is zero (for example, when R is PID or one dimensional Noetherian domain) and let M be anR-module. Then M axR(M) is a Noetherian topological space.

Proof. Let V_M^m(N) be a closed subset of M axR(M) for some submodule N of M. If V_M^m(N) is infinite, then (N : M) is contained in an infinite number of maximal ideals of R. Since the intersection of every infinite collection of maximal ideals ofR is zero, (N : M) = (0) so that V_M^m(N) = M axR(M). It follows that τ_M^m ⊆ τ^{f c} and hence M axR(M) is a Noetherian topological space because every

cofinite topological space is Noetherian.

Notation 3.3. Let M be an R-module and W be a subset of M axR(M). We will denote the intersection of all elements inW by=(W) and the closure ofW in M axR(M) (resp. SpecR(M)) byCl^m(W) (resp. Cl(W)).

Lemma 3.4. Let M be an R-module and W be a subset of M axR(M). Then Cl^m(W) =V_M^m(=(W)). Hence, W is closed if and only ifV_M^m(=(W)) =W. Proof. LetW be a subset ofM axR(M). It is well known that

Cl^m(W) =Cl(W)∩M axR(M).

But Cl(W) = V(=(W)) by [11, Proposition 5.1]. It follows that Cl^m(W) =

V_M^m(=(W)).

For a proper idealI of R, we recall that the J-radical I, denoted by J_R^m(I), is the intersection of all maximal ideals containingI. An ideal I of R is a J-radical ideal ifI=J_R^m(I).

Definition 3.5. LetM be anR-module. The J-radical of a submoduleN of M, denoted by J_M^m(N), is the intersection of all members of V_M^m(N). In case that V_M^m(N) =∅, we defineJ_M^m(N) =M. A submoduleN ofM is said to be a J-radical submodule ifN =J_M^m(N).

Theorem 3.6. Let M be an R-module. Then the following are equivalent.

(a) M axR(M)is a Noetherian topological space.

(b) The ascending chain condition for J-radical submodules ofM holds.

Proof. (a)⇒(b) Straightforward.

(b)⇒(a) Let

V_M^m(N₁)⊇V_M^m(N₂)⊇ · · · ⊇V_M^m(N_i)⊇ · · ·

be a descending chain of closed setsV_M^m(Ni) ofM axR(M), whereNiis a submodule ofM. Hence

J_M^m(N1)⊆J_M^m(N2)⊆ · · · ⊆J_M^m(Ni)⊆ · · ·

is an ascending chain of J-radical submodules ofM. So by hypothesis, there exists a k ∈ N such that for all n > k, we have J_M^m(N_k+n) = J_M^m(N_k). Now by using Lemma 3.4, for alln > k,V_M^m(N_k+n) =V_M^m(N_k) and the proof is completed.

Corollary 3.7. LetM be a NoetherianR-module. ThenM ax_R(M) is a Noether- ian topological space.

We recall that ifI is an ideal ofR, then the J-components of Iare the minimal members of the family of J-radical prime ideals containingI (see [16, p. 631]).

(5)

Definition 3.8. LetM be an R-module andL a submodule ofM. A submodule P of M is a J-component of L, if (P :M) is a J-component of (L:M). Clearly, this definition is the generalization of J-component of an ideal in rings.

Definition 3.9. A moduleM is said to have property (JFC) if every closed subset ofM axR(M) has a finite number of irreducible components.

Example 3.10. LetM be an R-module. Then M has property (JFC) in each of the following cases:

(a) M axR(M) is a Noetherian topological space (see parts (b) and (c) of Re- mark 2.2);

(b) Ris PID (see Proposition 3.2 and part (a));

(c) M is Noetherian (see Corollary 3.7 and part (a));

(d) M is semi local (see Remark 2.2 (e) and part (a)).

When M is the R-module R, then R has property (JFC) if and only if every ideal ofR has a finite number of J-components (see [16, p. 632]). Theorem 3.13 (d) extends the this property for modules.

The proof of the following lemma is easy and is omitted.

Lemma 3.11. LetM be aM ax-surjectiveR-module. Then the following hold.

(a) IfN is a submodule ofM, then

J_R^m((N :M)) = (J_M^m(N) :M).

(b) If q is a J-radical ideal of R containing AnnR(M), then there exists a submoduleQofM such that (Q:M) =q.

Remark 3.12. If S is a commutative ring with non zero identity, then there exists a one-to-one correspondence between the J-radical prime ideals of ringSand irreducible closed subsets ofM ax(S) (see [16, p. 631]).

Theorem 3.13. Let M be aM ax-surjectiveR-module. Then the following hold.

(a) If Y ⊆M axR(M), then Y is an irreducible closed subset of M axR(M) if and only ifY =V_M^m(N)for some submodule N of M such that(N :M)is a J-radical prime ideal ofR.

(b) If W ⊆ M ax_R(M) and L is submodule of M, then W is an irreducible component of V_M^m(L) if and only if W =V_M^m(N⁰) for some J-component N⁰ of L.

(c) IfZ⊆M axR(M), thenZ is an irreducible component of M axR(M)if and only ifZ=V_M^m(pM)for some J-component idealpofAnnR(M).

(d) M has property (JFC) if and only if every submodule of M has a finite number of J-components.

Proof. (a) (⇒) Let Y be an irreducible closed subset of M ax_R(M). Since Y is closed, Y =V_M^m(N) for some submoduleN of M. It turns out thatφ(V_M^m(N)) = V_R_¯^m((N :M)) is an irreducible closed subset ofM ax( ¯R) by Lemma 3.1 and Remark 2.2 (a). Now by Remark 3.12, (N :M) is a J-radical prime ideal of ¯R so that (N :M) is a J-radical prime ideal of R. Conversely, letV_M^m(K) be a closed subset of M ax_R(M), where K is a submodule of M such that (K : M) is a J-radical prime ideal ofR. We show thatV_M^m(K) is irreducible. To see this, letEandE⁰ be submodules ofM with

V_M^m(K)⊆V_M^m(E)∪V_M^m(E⁰).

(6)

Hence as in the proof of Lemma 3.1 (b), we have

VR¯^m((K:M))⊆VR¯^m((E:M))∪VR¯^m((E⁰ :M)).

Since (K:M) is a J-radical prime ideal ofR, it is easy to check that (K:M) is a J-radical prime ideal of ¯R. ThereforeV_R^m_¯ ((K:M)) is an irreducible closed subset ofM ax( ¯R) by Remark 3.12. Hence by Definition 2.1 (d),

V_R_¯^m((K:M))⊆V_R_¯^m((E:M))or V_R_¯^m((K:M))⊆V_R_¯^m((E⁰:M)).

Suppose thatV_R^m_¯ ((K:M))⊆V_R_¯^m((E:M)). This implies thatV_M^m(K)⊆V_M^m(E).

By similar arguments,V_M^m(K)⊆V_M^m(E⁰) whenV_R^m_¯ ((K:M))⊆V_R_¯^m((E⁰:M)).

(b) (⇒) LetW be an irreducible component ofV_M^m(L). By Definition 2.1 (e) and Remark 2.2 (g), W is an irreducible closed subset ofM axR(M). So by part (a),W =V_M^m(N₁⁰) for some submoduleN₁⁰ ofM such that (N₁⁰ :M) is a J-radical prime ideal ofR. We claim thatN₁⁰ is a J-component ofLor equivalently, (N₁⁰ :M) is a J-component of (L :M). Clearly (N₁⁰ : M)⊇(L: M) by using Lemma 3.11 (a). So by the above arguments, it is enough to show that (N₁⁰ :M) is a minimal member of the family of J-radical prime ideals containing (L:M). To see this, let qbe a J-radical prime ideal ofRwith

(L:M)⊆q⊆(N₁⁰ :M).

SinceM isM ax-surjective, there exists a submoduleQofM such thatq= (Q:M) by Lemma 3.11 (b). Hence

V_M^m(L)⊇V_M^m(Q)⊇V_M^m(N₁⁰).

Also V_M^m(Q) is an irreducible closed subset of V_M^m(L) by part (a), and Remark 2.2 (g). Since W =V_M^m(N₁⁰) is an irreducible component of V_M^m(L), by the above arguments, we haveV_M^m(Q) =V_M^m(N₁⁰). Now by using Lemma 3.11 (a), q= (N₁⁰ : M) as desired.

(⇐) Let N₂⁰⁰ be a J-component of L. Then V_M^m(N₂⁰⁰) is an irreducible closed subset ofV_M^m(L) by part (a) and Remark 2.2 (g). LetL⁰be a submodule ofM such that (L⁰ :M) is a J-radical prime ideal ofR and

V_M^m(N₂⁰⁰)⊆V_M^m(L⁰)⊆V_M^m(L).

Since N₂⁰⁰ be a J-component of L, by using Lemma 3.11 (a), we have V_M^m(N₂⁰⁰) = V_M^m(L⁰) as required.

(c) This follows from part (b) and Lemma 3.11 (b) and the fact that if N is a submodule ofM, then

V_M^m((N :M)M) =V_M^m(N).

(d) Follows from part (b).

LetX be a topological space. We consider strictly decreasing chainZ0, Z1,..., Zr of lengthrof irreducible closed subsetsZi ofX. The supremum of the lengths, taken over all such chains, is called the combinatorial dimension ofX and denoted bydim(X). For the empty set,∅, the combinatorial dimension of∅is defined to be

−1.

Theorem 3.14. Let M be aM ax-surjectiveR-module. Then the following hold.

(a) M ax_R(M) is a Noetherian topological space if and only if M ax( ¯R) is a Noetherian topological space.

(7)

(b) M axR(M)is a connected topological space if and only ifM ax( ¯R)is a connected topological space.

(c) M axR(M) is an irreducible topological space if and only if M ax( ¯R)is an irreducible topological space.

(d) M ax_R(M)is a quasi compact topological space if and only ifM ax( ¯R)is a quasi compact topological space.

(e) dim(M ax_R(M)) =dim(M ax( ¯R)).

Proof. Letφ:M ax_R(M)→M ax( ¯R) be the natural map of M ax_R(M).

(a) (⇒) Let V_R_¯^m( ¯I₁) ⊇ V_R_¯^m( ¯I₂) ⊇ ... ⊇V_R^m_¯ ( ¯I_i) ⊇ ... be a descending chain of closed sets in M ax( ¯R), where each ¯Ii is an ideal of ¯R. Since φ is continuous by Lemma 3.1 (a),

φ⁻¹(V_R_¯^m( ¯I₁))⊇φ⁻¹(V_R_¯^m( ¯I₂))⊇...⊇φ⁻¹(V_R_¯^m( ¯I_i))⊇...

is a descending chain of closed sets in M axR(M). By hypothesis, there exists a t∈Nsuch that for all n > t,φ⁻¹(V_R_¯^m( ¯It+n)) =φ⁻¹(V_R^m_¯ ( ¯It)). Hence for all n > t, we have V_R_¯^m( ¯I_t+n) = V_R_¯^m( ¯I_t) because φ is surjective. Therefore, M ax( ¯R) is a Noetherian topological space. To show the converse, by Theorem 3.6, it is enough to show that the ascending chain condition for J-radical submodules of M holds.

To see this, let

N₁⊆N₂⊆ · · · ⊆N_i⊆ · · ·

be an ascending chain of J-radical submodules ofM. Then by Lemma 3.11 (a), one can see that

(N1:M)⊆(N2:M)⊆ · · · ⊆(Ni:M)⊆ · · ·

is an ascending chain of J-radical ideals of ¯R. So by Theorem 3.6, there exists a k∈Nsuch that for alln > k, (Nk+n:M) = (Nk :M).Hence for alln > k,

V_M^m(Nk+n) =V_M^m((Nk+n:M)M) =V_M^m((Nk :M)M) =V_M^m(Nk).

So for alln > k, we have

Nk+n =J_M^m(Nk+n) =J_M^m(Nk) =Nk, as desired.

(b) First assume thatM ax_R(M) is a connected topological space. ThenM ax( ¯R) = φ(M ax_R(M)) is connected by Lemma 3.1 and Remark 2.2 (a). To see the reverse implication, we assume thatM ax( ¯R) is a connected topological space. IfM axR(M) is a disconnected topological space, then there exist submodules N and K of M such that

M ax_R(M) =V_M^m(N)∪V_M^m(K) and

V_M^m(N)∩V_M^m(K) =∅,

where V_M^m(N)6=∅, andV_M^m(K)6=∅. Hence as in the proof of Lemma 3.1 (b), we have

M ax( ¯R) =VR¯^m((N:M))∪VR¯^m((K:M)).

It is easy to cheek that

V_R_¯^m((N :M))∩V_R_¯^m((K:M)) =∅, V_R_¯^m((N :M))6=∅, and V_R_¯^m((K:M))6=∅.

Therefore M ax( ¯R) is a disconnected topological space, a contradiction. Hence M ax_R(M) is a connected topological space.

(c) We have similar argument as in part (b).

(8)

(d) (⇒) This follows from Lemma 3.1 (a) and Remark 2.2 (a). To show the converse, let {V_M^m(Nα) : α ∈ Λ} be a family of closed subset of M axR(M) such that ∩α∈ΛV_M^m(Nα) = ∅, where Nα is a submodule of M for every α∈ Λ. Then {φ(V_M^m(N_α)) :α∈Λ} is a family of closed subset of M ax( ¯R) becauseφis closed by Lemma 3.1 (b). Sinceφis surjective, it is easy to see that ∩α∈Λφ(V_M^m(N_α)) =

∅. As M ax( ¯R) is quasi compact, there exists a finite subset Γ of Λ such that

∩_α∈Γφ(V_M^m(N_α)) = ∅. This implies that∩_α∈ΓV_M^m(N_α) =∅ and hence M ax_R(M) is quasi compact.

(e) LetZ0⊃Z1⊃...⊃Zn be a descending chain of irreducible closed subset of M axR(M). Then by Theorem 3.13 (a), for i(1 ≤i≤n), there exists submodule Li of M such that (Li :M) is a J-radical prime ideal ofR andZi =V_M^m(Li). It follows that

VR¯^m((L0:M))⊃VR¯^m((L1:M))...⊃VR¯^m((Ln:M))

is a descending chain of irreducible closed subset ofM ax( ¯R) by Remark 3.12. Hence dim(M ax_R(M))≤dim(M ax( ¯R)).

Now let

A₀⊃A₁⊃...⊃A_t

be a descending chain of irreducible closed subset ofM ax( ¯R). By Remark 3.12, for eachi(1≤i≤t), there exists a J-radical prime ideal ¯p_iof ¯Rsuch thatA_i =V_R^m_¯ (¯p_i).

This yields that

p₀⊂p₁⊂...⊂p_t

is an ascending chain of J-radical prime ideal ofR. SinceM isM ax-surjective, by Lemma 3.11 (b), for every p_i (1≤i≤t), there exists a submodule Q_i of M such thatp_i= (Q_i :M). Hence by Theorem 3.13 (a),

V_M^m(Q₀)⊃V_M^m(Q₁)⊃...⊃V_M^m(Q_t)

is a descending chain of irreducible closed subset of M axR(M). It follows that dim(M axR(M))≥dim(M ax( ¯R)) and the proof is completed.

Corollary 3.15. LetM be aM ax-surjectiveR-module. Then the following hold.

(a) IfRis Noetherian, then M axR(M) is a Noetherian topological space.

(b) If Ψ is the family of all J-radical prime ideal ofR, then we have dim(M axR(M)) =sup{n|p0⊂p1⊂...⊂pn is an ascending chain of Ψ}.

Proof. (a) Follows from Theorem 3.14 (a).

(b) Apply the technique of Theorem 3.14 (e).

Remark 3.16. We recall that anR-moduleM is a Hilbert module if every prime submodule inM is the intersection of all the maximal submodules containing it.

For example, every finitely generated divisible module over an integral domain is a Hilbert module (see [13, p. 2]). LetM be a Hilbert R-module. IfM axR(M) is connected (resp. irreducible) topological space, thenSpec_R(M) is connected (resp.

irreducible) topological space. Since if M is Hilbert, by [11, Proposition 5.1] it is easy to see that Cl(M ax_R(M)) =Spec_R(M). Now the result follows from the Remark 2.2 (f).

Definition 3.17. We say that a topological spaceW is aM ax-spectral space ifW is homeomorphic with the maximal ideal space of some ringS (with the topology inherited fromSpec(S)).

(9)

Remark 3.18. M ax-spectral spaces have been characterized by Hochster [8, p.57, Proposition 11] as the topological spaces W which satisfy the following conditions:

(a) W is a T1space;

(b) W is quasi-compact.

Proposition 3.19. LetM be anR-module. Then the following are equivalent.

(a) M isM ax-injective.

(b) M ax_R(M) is a T₀ space.

(c) M ax_R(M) is a T₁ space.

(d) M axR(M) is a T2 space.

Proof. Straightforward.

Corollary 3.20. LetM be anR-module.

(a) IfM ax_R(M) is aM ax-spectral topological space, thenM isM ax-injective.

(b) IfM is primeful andSpecR(M) is aM ax-spectral topological space, then SpecR(M) =M axR(M).

Proof. This follows from Remark 3.18, Proposition 3.19, and [2, Theorem 4.3].

We recall that a topological spaceXis spectral if it is homeomorphic toSpec(S) with the Zariski topology for some ringS (see [8]).

Remark 3.21. Let M = Z(p^∞)⊕Z. Then M is a primeful Z-module and SpecR(M) is a spectral topological space butSpecR(M)6=M axR(M) by [3, Table of examples 3.1 and 3.2]. This shows that part (b) in Corollary 3.20 is not valid in general if the word ”M ax-spectral” is replaced with ”spectral”.

Example 3.22.

(a) M ax_Z(Z2⊕Z3) is aM ax-spectral topological space by [3, Table of examples 3.1 and 3.2] and Remark 3.18.

(b) M ax_Q(Q⊕Q) is not aM ax-spectral topological space because 0⊕Qand Q⊕0 are maximal submodules of the Q-module Q⊕Q with (0⊕Q : Q⊕Q) = (Q⊕0 :Q⊕Q), while 0⊕Q6=Q⊕0. ThusM ax_Q(Q⊕Q) is not M ax-spectral by Corollary 3.20 (a).

Let M be an R-module such that M axR(M) is a M ax-spectral topological space. For a submodule N of M, it is natural to ask the following question: Is M axR(M/N) aM ax-spectral topological space?

In Proposition 3.23 (c), we give a positive answer to this question under some additional conditions.

Proposition 3.23. LetM be anRmodule and letN be a submodule ofM. Then the following hold.

(a) IfM ax_R(M) is a T₁ topological space, then so isM ax_R(M/N).

(b) IfM axR(M) is a Noetherian topological space, then so isM axR(M/N).

(c) Let M axR(M) be a M ax-spectral space. Then M axR(M/N) is aM ax- spectral space in the following cases:

(i) The subspace H:={Q∈M axR(M)|Q⊇N}ofM axR(M) is closed;

(ii) Ris a ring such that the intersection of every infinite collection of maximal ideals ofRis zero (for example, whenRis PID or one dimensional Noetherian domain).

(10)

Proof. (a) Follows from Proposition 3.19 and the fact that ifN is a submodule of M, then

M axR(M/N) ={Q/N|Q∈M axR(M), Q⊇N}.

(b) We define the mapf :M axR(M/N)→H, whereH :={Q∈M axR(M)|Q⊇ N}andf(Q/N) =Qfor everyQ/N ∈M axR(M/N). Clearlyf is a bijection map.

Now letV_M^m(E)∩H be a closed set ofH, whereE is a submodule ofM. Then f⁻¹(V_M^m(E)∩H) =f⁻¹(V_M^m(E))∩f⁻¹(H) =f⁻¹(V_M^m(E))∩M ax_R(M/N)

=f⁻¹(V_M^m(E)) =V_M^m(K/N),

where K= (E :M)M+N. Sof :M axR(M/N)→H is a continuous map. It is easy to check that

f(V_M^m(L/N)) =V_M^m(L)∩H

for every submoduleLofM containingN. Hencef :M axR(M/N)→H is a closed map so thatM axR(M/N) is homeomorphic withH. Now sinceM axR(M) is Noe- therian,H is Noetherian by Remak 2.2 (b). Hence M axR(M/N) is a Noetherian space as desired.

(c)(i) As in the proof part (b), we see thatM ax_R(M/N) is homeomorphic with H. Now the result follows by part (a), Remark 3.18, and Remark 2.2 (d).

(c)(ii) This follows from Proposition 3.2, Remark 3.18, Remark 2.2 (b), and part

(a).

The next theorem is an important result about an R-module M for which M axR(M) is M ax-spectral. This result is obtained by combining Lemma 3.1, Proposition 3.2, Theorem 3.6, Proposition 3.19, Remark 2.2 (e), and Remark 3.18.

Theorem 3.24. LetM be aM ax-injectiveR-module. ThenM axR(M)is aM ax- spectral topological space in each of the following cases:

(a) M isM ax-surjective;

(b) Im(φ) is quasi compact, where φ : M ax_R(M) → M ax( ¯R) is the natural map ofM ax_R(M);

(c) Ann_R(M)is a maximal ideal ofR;

(d) M ax_R(M)is a finite set;

(e) M ax(R)is a finite set;

(f) M ax( ¯R)is Noetherian, in particular whenR is Noetherian;

(g) The intersection of every infinite of maximal ideals ofR is zero, in particular whenR is PID or one dimensional Noetherian domain;

(h) The ascending chain condition for J-radical submodules ofM holds.

Remark 3.25. Let M =⊕_i∈_NZ/p_iZ, wherep_i is a prime integer for each i∈N. ThenM is anX-injective module overZbutSpec_Z(M) is not a spectral topological space by [3, Table of examples 3.1 and 3.2]. This shows that the words ”M ax- injective”, ”M axR(M)”, and ”M ax-spectral” in part (g) of Theorem 3.24, can not be replaced with ”X-injective”, ”SpecR(M)”, and ”spectral” respectively.

AnR-moduleM is multiplication if for every submoduleN ofM, there exits an idealIofR such thatN =IM (see [14]).

Corollary 3.26. Let M be an R-module. Then M axR(M) is a M ax-spectral topological space in each of the following cases:

(a) M is finitely generated and multiplication;

(11)

(b) M is primeful and top; (We refer the reader to [14] and [5] for the concept and properties of top modules.

(c) M is primeful andX-injective;

(d) M isX-injective andR is PID.

Proof. This follows from parts (a) and (g) of Theorem 3.24 and taking into account the following facts from [14, Theorem 3.5], [2, Proposition 3.3], [12, Theorem 2.2 3.3], and [1, Proposition 3.3 (c)],

Fact 1. Let denote the class of multiplication, top, X-injective, and M ax- injective modules respectively by Γ1, Γ2, Γ3, and Γ4, then

Γ1⊆Γ2⊆Γ3⊆Γ4.

Fact 2. If we denote the class of finitely generated, primeful, andM ax-surjective modules respectively by Ω1, Ω2, and Ω3, then

Ω₁⊆Ω₂⊆Ω₃.

In below, by using Corollary 3.26 (d) and [3, Table of examples 3.1], we provide further examples about of modules such that their maximal submodules areM ax- spectral.

Example 3.27. Letp(resp. p_i,i∈N) be a prime number. Then for each of the following cases,M ax_Z(M) is aM ax-spectral topological space.

(a) M =Z(p)=S⁻¹Z, where S=Z\(p).

(b) M =Z(p^∞)⊕Z. (c) M =⊕_i∈NZ/piZ. (d) M =Q

i∈NZ/piZ. (e) M =Q⊕(⊕_i∈NZ/piZ).

Unfortunately, we have not bee able to find a M ax-injective R-moduleM such thatM axR(M) is notM ax-spectral. This motivates the following question.

Question 3.28. Let M be a M ax-injective R-module. Is M ax_R(M) a M ax- spectral topological space?

Acknowledgments. The authors would like to thank Dr. R. Ovlyaee-Sarmazdeh for several helpful conversations in this work. Also we are grateful to the referee for careful reading of the manuscript.

References

[1] H. Ansari-Toroghy and R. Ovlyaee-Sarmazdeh,Modules for which the natural map of the maximal spectrum is surjective, Colloq. Math,119(2010), 217–227.

[2] ,On the prime spectrum ofX-injective modules, Comm. Algebra,38(2010), 2606–

2621.

[3] ,On the prime spectrum of a module and Zariski topologies, Comm. Algebra,38 (2010), 4461-4475.

[4] H. Ansari-Toroghy and S. Keyvani,Some new classes of modules, (2011), submitted.

[5] H. Ansari-Toroghy and S. Keyvani,Strongly top modules, Bull. Malays. Math. Sci. Soc.

(2), accepted.

[6] M.F. Atiyah and I.G. Macdonald,Introduction to commutative algebra, Addison-Wesley, 1969.

[7] N. Bourbaki,Algebra Commutative, Chap. 1,2, Paris: Hermann, 1961.

(12)

[8] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc,142 (1969), 43–60.

[9] Chin-Pi Lu,Prime submodules of modules, Comment. Math. Univ. St. Pauli, 33(1984), no. 1, 61–69.

[10] ,Spectra of modules, Comm. Algebra,23(1995), no. 10, 3741–3752.

[11] ,The zariski topology on the prime spectrum of a module, Houston J. Math,25 (1999), no. 3, 417–432.

[12] ,A module whose prime spectrum has the surjective natural map, Houston J. Math, 33(1) (2007), 125–143.

[13] M. Maani Shirazi and H. Sharif,Hilbert modules, International Journal of Pure and Applied Mathematics,20(2005), no. 1, 1–7.

[14] R.L. McCasland, M.E. Moore, and P.F. Smith,On the spectrum of a module over a commutative ring, Comm. Algebra,25(1997), no. 1, 79–103.

[15] Jesper M. Moller,General Topology, Matematisk Institut, Universitetsparken 5, DK-2100 Kobenhavn.

[16] J. Ohm and R.L. Pendleton,Ring with Noetherian spectrum, Duke Math. J,35(1968), 631–639.

Department of pure Mathematics, Faculty of mathematical Science, University of Guilan, P. O. Box 41335-19141 Rasht, Iran.

E-mail address:[email protected]

Department of pure Mathematics, Faculty of mathematical Science, University of Guilan, P. O. Box 41335-19141 Rasht, Iran.

E-mail address:Siamak [email protected]