Home Zariski topology on the secondary-like spectrum of a module
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Zariski topology on the secondary-like spectrum of a module

  • Saif Salam and Khaldoun Al-Zoubi EMAIL logo
Published/Copyright: April 5, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

Let ℜ be a commutative ring with unity and ℑ be a left ℜ -module. We define the secondary-like spectrum of ℑ to be the set of all secondary submodules K of ℑ such that the annihilator of the socle of K is the radical of the annihilator of K , and we denote it by Spec L ( ℑ ) . In this study, we introduce a topology on Spec L ( ℑ ) having the Zariski topology on the second spectrum Spec s ( ℑ ) as a subspace topology and study several topological structures of this topology.

Keywords: secondary-like spectrum; Zariski topology; second spectrum
MSC 2010: 13C13; 13C99; 54B99

1 Introduction and preliminaries

Throughout this article, all rings are commutative with identity and all modules are left unital modules. Let ℑ be an ℜ -module. By I ◃ ℜ (resp. N ≤ ℑ ), we mean that I is an ideal of ℜ (resp. N is a submodule of ℑ ). Let ℑ be an ℜ -module, N ≤ ℑ , and I ◃ ℜ . The annihilator of N in ℜ (resp. the annihilator of I in ℑ ) will be expressed by Ann ℜ ( N ) = { r ∈ ℜ ∣ r N = { 0 } } (resp. Ann ℑ ( I ) = { m ∈ ℑ ∣ Im = { 0 } } ). For a ring ℜ , the set of all prime ideals of ℜ is denoted by Spec ( ℜ ) , and it is called the prime spectrum of ℜ . For any ideal I of ℜ , define V ℜ ( I ) as the set of all prime ideals containing I , i.e., V ℜ ( I ) = { p ∈ Spec ( ℜ ) ∣ p ⊇ I } . Then, there exists a topology on Spec ( ℜ ) having { V ℜ ( I ) ∣ I ◃ ℜ } as the collection of closed sets. This topology is known as the Zariski topology on Spec ( ℜ ) (see, for example, [1,2]).

Let ℑ be an ℜ -module. A submodule S of ℑ is called second if S ≠ { 0 } , and for every r ∈ ℜ , we have r S = S or r S = { 0 } . It is clear that Ann ℜ ( S ) ∈ Spec ( ℜ ) for any second submodule S of ℑ . The second spectrum of ℑ , given by Spec s ( ℑ ) , is the set of all second submodules of ℑ . If Spec s ( ℑ ) = ∅ , then we say that ℑ is a secondless ℜ -module. The socle of a submodule N of ℑ , expressed by sec ( N ) , is the sum of all second submodules of ℑ . If there are no second submodules contained in N , then sec ( N ) is defined to be { 0 } . The second submodules and the socle of submodules have been studied by many authors (see, for example, [3–6]). A submodule K of ℑ is said to be secondary if K ≠ { 0 } , and for every r ∈ ℜ , we have r K = K or there exists a positive integer n such that r n K = { 0 } . For any ideal I of ℜ , we denote the radical of I by I . It is easily seen that if K is a secondary submodule of ℑ , then Ann ℜ ( K ) is a primary ideal of ℜ , and hence, Ann ℜ ( K ) ∈ Spec ( ℜ ) . For more details concerning the secondary submodules, one can look in [7,8].

Let ℑ be an ℜ -module and let ζ s * ( ℑ ) = { V s * ( N ) ∣ N ≤ ℑ } , where V s * ( N ) = { S ∈ Spec s ( ℑ ) ∣ N ⊇ S } for any N ≤ ℑ . We say that ℑ is a cotop module if ζ s * ( ℑ ) is closed under finite unions, and in this case, ζ s * ( ℑ ) satisfies the axioms for closed sets of a topology on Spec s ( ℑ ) . The generated topology is called the quasi-Zariski topology on Spec s ( ℑ ) . Unlike ζ s * ( ℑ ) , ζ s ( ℑ ) = { V s ( N ) ∣ N ≤ ℑ } , where V s ( N ) = { S ∈ Spec s ( ℑ ) ∣ Ann ℜ ( S ) ⊇ Ann ℜ ( N ) } for any N ≤ ℑ always satisfies all of the topology axioms for the closed sets. The resulting topology is called the Zariski topology on Spec s ( ℑ ) . When Spec s ( ℑ ) ≠ ∅ , the map ψ : Spec s ( ℑ ) → Spec ( ℜ ∕ Ann ℜ ( ℑ ) ) defined by S ↦ Ann ℜ ( S ) ∕ Ann ℜ ( ℑ ) for every S ∈ Spec s ( ℑ ) is called the natural map of Spec s ( ℑ ) . For more information about the topologies on Spec s ( ℑ ) and the natural map of Spec s ( ℑ ) , see [9–12]. In addition, in [13,14], the Zariski topology on the graded second spectrum of graded modules has been studied.

In this work, we call the set of all secondary submodules K of an ℜ -module ℑ satisfying the condition Ann ℜ ( sec ( K ) ) = Ann ℜ ( K ) the secondary-like spectrum of ℑ , and it is denoted by Spec L ( ℑ ) . It is clear that Spec s ( ℑ ) ⊆ Spec L ( ℑ ) . But the converse inclusion is not true in general. For example, if ℑ is a non-zero vector space over a field F , then Spec s ( ℑ ) = Spec L ( ℑ ) = { N ≤ ℑ ∣ N ≠ { 0 } } . On the other hand, if we take Z 8 as Z 8 -module, then it is easy to see that the submodule K = { 0 , 2 , 4 , 6 } ∈ Spec L ( Z 8 ) . However, K = { 0 , 2 , 4 , 6 } ∉ Spec s ( Z 8 ) as 2 K = { 0 , 4 } ≠ N and 2 K ≠ { 0 } . Also, if we take Z as Z -module, then Spec s ( Z ) = Spec L ( Z ) = ∅ . For an ℜ -module ℑ , it is obvious that sec ( K ) ≠ { 0 } for any K ∈ Spec L ( ℑ ) as Ann ℜ ( sec ( K ) ) = Ann ℜ ( K ) ∈ Spec ( ℜ ) .

We start this work by introducing the notion of the secondary cotop module, which is a generalization of the cotop module. For this, we define the variety of any submodule N of an ℜ -module ℑ by ν s * ( N ) = { K ∈ Spec L ( ℑ ) ∣ sec ( K ) ⊆ N } and we set Θ s * ( ℑ ) = { ν s * ( N ) ∣ N ≤ ℑ } . Then, we say that ℑ is a secondary cotop module if Θ s * ( ℑ ) is closed under finite unions. It is clear that ν s * ( ℑ ) = Spec L ( ℑ ) , ν s * ( { 0 } ) = ∅ and ⋂ j ∈ Δ ν s * ( N j ) = ν s * ( ⋂ j ∈ Δ N j ) for any family of submodules { N j } j ∈ Δ and any non-empty index set Δ (Theorem 2.1). So if ℑ is a secondary cotop module, then there exists a topology on Spec L ( ℑ ) having Θ s * ( ℑ ) as a collection of closed sets, and this topology is called the quasi-Zariski topology on Spec L ( ℑ ) . Next, we introduce another variety for any N ≤ ℑ by ν s ( N ) = { K ∈ Spec L ( ℑ ) ∣ Ann ℜ ( N ) ⊆ Ann ℜ ( K ) } and prove that there always exists a topology on Spec L ( ℑ ) having Θ s ( ℑ ) = { ν s ( N ) ∣ N ≤ ℑ } as the collection of closed sets. We call this topology the Zariski topology on Spec L ( ℑ ) or simply Sℒ -topology (Theorem 2.3). We provide some relationships between the varieties ν s * ( N ) , ν s ( N ) , V s * ( N ) , and V s ( N ) for any submodule N of an ℜ -module ℑ and conclude that every secondary cotop module is a cotop module (Lemma 2.4 and Corollary 2.5). In addition, using these relations, we see that the Zariski topology on Spec s ( ℑ ) for an ℜ -module ℑ is a topological subspace of Spec L ( ℑ ) equipped with the Sℒ -topology. Next, for an ℜ -module ℑ , we introduce the map φ : Spec L ( ℑ ) → Spec ( ℜ ∕ Ann ℜ ( ℑ ) ) by φ ( K ) = Ann ℜ ( K ) ∕ Ann ℜ ( ℑ ) to obtain some relations between the properties of Spec L ( ℑ ) and Spec ( ℜ ∕ Ann ℜ ( ℑ ) ) . For example, we relate the connectedness of Spec L ( ℑ ) to the connectedness of both Spec ( ℜ ∕ Ann ℜ ( ℑ ) ) and Spec s ( ℑ ) using φ and ψ (Theorem 2.12). In Section 2, among other results, we show that the secondary-like spectrums of two modules are homeomorphic if there exists a module isomorphism between these modules (Corollary 2.15). In Section 3, we introduce a base for the Zariski topology on Spec L ( ℑ ) (Theorem 3.1). Also, we prove that the surjectivity of φ implies that the basic open sets are quasi-compact and conclude that Spec L ( ℑ ) is quasi-compact (Theorem 3.4). In Section 4, we investigate the irreducibility of Spec L ( ℑ ) with respect to the Sℒ -topology. In particular, it is shown that if φ is surjective, then every irreducible closed subset of Spec L ( ℑ ) has a generic point, and we determine exactly the set of all irreducible components of Spec L ( ℑ ) (Theorem 4.5 and Corollary 4.8). Furthermore, we study Spec L ( ℑ ) from the viewpoint of being a T 0 -space, a T 1 -space, and a spectral space.

2 Zariski topology on Spec L ( ℑ )

In this section, we define two varieties of submodules and we use their properties to construct the quasi-Zariski topology and the Sℒ -topology on Spec L ( ℑ ) . In addition, we introduce some relationships between Spec L ( ℑ ) , Spec ( ℜ ∕ Ann ℜ ( ℑ ) ) , and Spec s ( ℑ ) .

Theorem 2.1

Let ℑ be an ℜ -module. Then, we have the following:

  1. ν s * ( ℑ ) = Spec L ( ℑ ) and ν s * ( { 0 } ) = ∅ .

  2. ⋂ j ∈ Δ ν s * ( N j ) = ν s * ( ⋂ j ∈ Δ N j ) for any family of submodules { N j } j ∈ Δ and any non-empty index set Δ .

  3. ν s * ( N ) ∪ ν s * ( L ) ⊆ ν s * ( N + L ) for any N , L ≤ ℑ .

  4. If N 1 , N 2 ≤ ℑ with N 1 ⊆ N 2 , then ν s * ( N 1 ) ⊆ ν s * ( N 2 ) .

  5. ν s * ( sec ( N ) ) = ν s * ( N ) for any submodule N of ℑ .

Proof

The proof is obvious.□

The reverse inclusion in Theorem 2.1(3) is not true in general. For example, take Im = Z 2 × Z 2 as Z -module and let N = Z 2 × { 0 } and L = { 0 } × Z 2 . Then, ν s * ( N + L ) = ν s * ( ℑ ) = Spec L ( ℑ ) . Note that ℑ ∈ Spec s ( ℑ ) ⊆ Spec L ( ℑ ) . It follows that ℑ ∈ ν s * ( N + L ) and sec ( ℑ ) = ℑ . But ℑ ⊈ N and ℑ ⊈ L . Therefore, ℑ ∉ ν s * ( N ) ∪ ν s * ( L ) .

Recall that an ℜ -module ℑ is said to be comultiplication module if any submodule N of ℑ has the form Ann ℑ ( I ) for some ideal I of ℜ . By [15, Lemma 3.7], if N is a submodule of a comultiplication ℜ -module ℑ , then N = Ann ℑ ( Ann ℜ ( N ) ) .

Theorem 2.2

Every comultiplication module is a secondary cotop module.

Proof

Let ℑ be a comultiplication ℜ -module and N 1 , N 2 ≤ ℑ . It is sufficient to show that ν s * ( N 1 + N 2 ) ⊆ ν s * ( N 1 ) + ν s * ( N 2 ) . So let K ∈ ν s * ( N 1 + N 2 ) . Then, sec ( K ) ⊆ N 1 + N 2 . Hence, Ann ℜ ( N 1 ) ∩ Ann ℜ ( N 2 ) = Ann ℜ ( N 1 + N 2 ) ⊆ Ann ℜ ( sec ( K ) ) = Ann ℜ ( K ) ∈ Spec ( ℜ ) as K is a secondary submodule. This implies that Ann ℜ ( N 1 ) ⊆ Ann ℜ ( sec ( K ) ) or Ann ℜ ( N 2 ) ⊆ Ann ℜ ( sec ( K ) ) . Thus, sec ( K ) ⊆ Ann ℑ ( Ann ℜ ( sec ( K ) ) ) ⊆ Ann ℑ ( Ann ℜ ( N 1 ) ) = N 1 or

sec ( K ) ⊆ Ann ℑ ( Ann ℜ ( sec ( K ) ) ) ⊆ Ann ℑ ( Ann ℜ ( N 2 ) ) = N 2 .

Therefore, K ∈ ν s * ( N 1 ) ∪ ν s * ( N 2 ) .□

In the following theorem, we construct the Zariski topology on Spec L ( ℑ ) of an ℜ -module ℑ by introducing another variety of any submodule N of ℑ by ν s ( N ) = { K ∈ Spec L ( ℑ ) ∣ Ann ℜ ( N ) ⊆ Ann ℜ ( K ) } .

Theorem 2.3

The following hold for any ℜ -module ℑ :

  1. ν s ( ℑ ) = Spec L ( ℑ ) and ν s ( { 0 } ) = ∅ .

  2. ⋂ i ∈ I ν s ( N i ) = ν s ( ⋂ i ∈ I Ann ℑ ( Ann ℜ ( N i ) ) ) for any family of submodules { N i } i ∈ I ≠ ∅ .

  3. ν s ( N 1 ) ∪ ν s ( N 2 ) = ν s ( N 1 + N 2 ) for any N 1 , N 2 ≤ ℑ .

Proof

(1) is straightforward.

(2) Let K ∈ ⋂ i ∈ I ν s ( N i ) . Then, Ann ℜ ( N i ) ⊆ Ann ℜ ( K ) , and thus,

Ann ℑ ( Ann ℜ ( K ) ) ⊆ Ann ℑ ( Ann ℜ ( N i ) ) ,

for each i ∈ I ≠ ∅ . Hence, Ann ℑ ( Ann ℜ ( K ) ) ⊆ ⋂ i ∈ I Ann ℑ ( Ann ℜ ( N i ) ) , which implies that Ann ℜ ( ⋂ i ∈ I Ann ℑ ( Ann ℜ ( N i ) ) ) ⊆ Ann ℜ ( Ann ℑ ( Ann ℜ ( K ) ) ) = Ann ℜ ( Ann ℑ ( Ann ℜ ( sec ( K ) ) ) ) = Ann ℜ ( sec ( K ) ) = Ann ℜ ( K ) . So K ∈ ν s ( ⋂ i ∈ I Ann ℑ ( Ann ℜ ( N i ) ) ) and so ⋂ i ∈ I ν s ( N i ) ⊆ ν s ( ⋂ i ∈ I Ann ℑ ( Ann ℜ ( N i ) ) ) . Conversely, let K ∈ ν s ( ⋂ i ∈ I Ann ℑ ( Ann ℜ ( N i ) ) ) . Then, we have

Ann ℜ ( ⋂ i ∈ I Ann ℑ ( Ann ℜ ( N i ) ) ) ⊆ Ann ℜ ( K ) .

But for any i ∈ I ≠ ∅ , ⋂ i ∈ I Ann ℑ ( Ann ℜ ( N i ) ) ⊆ Ann ℑ ( Ann ℜ ( N i ) ) , and thus, Ann ℜ ( N i ) = Ann ℜ ( Ann ℑ ( Ann ℜ ( N i ) ) ) ⊆ Ann ℜ ( ⋂ i ∈ I Ann ℑ ( Ann ℜ ( N i ) ) ) ⊆ Ann ℜ ( K ) . Therefore, K ∈ ⋂ i ∈ I ν s ( N i ) , as desired.

(3) Since N 1 ⊆ N 1 + N 2 and N 2 ⊆ N 1 + N 2 , we have ν s ( N 1 ) ⊆ ν s ( N 1 + N 2 ) and ν s ( N 2 ) ⊆ ν s ( N 1 + N 2 ) . Therefore ν s ( N 1 ) ∪ ν s ( N 2 ) ⊆ ν s ( N 1 + N 2 ) . Conversely, let K ∈ ν s ( N 1 + N 2 ) . Then, Ann ℜ ( N 1 ) ∩ Ann ℜ ( N 2 ) = Ann ℜ ( N 1 + N 2 ) ⊆ Ann ℜ ( K ) ∈ Spec ( ℜ ) , and hence, Ann ℜ ( N 1 ) ⊆ Ann ℜ ( K ) or Ann ℜ ( N 2 ) ⊆ Ann ℜ ( K ) . This implies that K ∈ ν s ( N 1 ) ∪ ν s ( N 2 ) .□

In the following lemma, we state some useful relationships between the varieties V s * ( N ) , V s ( N ) , ν s * ( N ) , and ν s ( N ) .

Lemma 2.4

Let N and N ′ be the submodules of an ℜ -module ℑ and I be an ideal of ℜ . Then, the following hold:

  1. V s ( N ) = ν s ( N ) ∩ Spec s ( ℑ ) .

  2. V s * ( N ) = ν s * ( N ) ∩ Spec s ( ℑ ) .

  3. If Ann ℜ ( N ) = Ann ℜ ( N ′ ) , then ν s ( N ) = ν s ( N ′ ) . The converse is also true if N , N ′ ∈ Spec L ( ℑ ) .

  4. ν s * ( N ) ⊆ ν s ( N ) . Furthermore, if ℑ is a comultiplication module, then the equality holds.

  5. ν s ( Ann ℑ ( I ) ) = ν s ( Ann ℑ ( I ) ) = ν s * ( Ann ℑ ( I ) ) = ν s * ( Ann ℑ ( I ) ) .

  6. ν s ( N ) = ν s ( Ann ℑ ( Ann ℜ ( N ) ) ) = ν s ( Ann ℑ ( Ann ℜ ( N ) ) ) = ν s * ( Ann ℑ ( Ann ℜ ( N ) ) ) = ν s * ( Ann ℑ ( Ann ℜ ( N ) ) ) .

  7. If N ∈ Spec L ( ℑ ) or ℑ is a comultiplication module, then ν s ( N ) = ν s ( sec ( N ) ) .

Proof

(1), (2), (3), and (4) are clear.

(5) We first show that ν s ( Ann ℑ ( I ) ) = ν s * ( Ann ℑ ( I ) ) for any ideal I of ℜ . So let I be an ideal of ℜ . By (4), ν s * ( Ann ℑ ( I ) ) ⊆ ν s ( Ann ℑ ( I ) ) . For the reverse inclusion, let K ∈ ν s ( Ann ℑ ( I ) ) . Then, Ann ℜ ( Ann ℑ ( I ) ) ⊆ Ann ℜ ( K ) = Ann ℜ ( sec ( K ) ) , and thus, sec ( K ) ⊆ Ann ℑ ( Ann ℜ ( sec ( K ) ) ) ⊆ Ann ℑ ( Ann ℜ ( Ann ℑ ( I ) ) ) = Ann ℑ ( I ) , which implies that K ∈ ν s * ( Ann ℑ ( I ) ) . Therefore, ν s ( Ann ℑ ( I ) ) ⊆ ν s * ( Ann ℑ ( I ) ) . Now, it is sufficient to prove that ν s * ( Ann ℑ ( I ) ) = ν s * ( Ann ℑ ( I ) ) . Since I ⊆ I , we have Ann ℑ ( I ) ⊆ Ann ℑ ( I ) , which implies that ν s * ( Ann ℑ ( I ) ) ⊆ ν s * ( Ann ℑ ( I ) ) by Theorem 2.1(4). Conversely, let K ∈ ν s * ( Ann ℑ ( I ) ) . Then, we have sec ( K ) ⊆ Ann ℑ ( I ) , and hence, we obtain I ⊆ Ann ℜ ( Ann ℑ ( I ) ) ⊆ Ann ℜ ( sec ( K ) ) = Ann ℜ ( K ) , which implies that I ⊆ Ann ℜ ( sec ( K ) ) . Thus, sec ( K ) ⊆ Ann ℑ ( Ann ℜ ( sec ( K ) ) ) ⊆ Ann ℑ ( I ) . Therefore, K ∈ ν s * ( Ann ℑ ( I ) ) , as needed.

(6) Note that for any K ∈ Spec L ( ℑ ) , we have K ∈ ν s ( N ) ⇔

Ann ℜ ( Ann ℑ ( Ann ℜ ( N ) ) ) = Ann ℜ ( N ) ⊆ Ann ℜ ( K ) ⇔

K ∈ ν s ( Ann ℑ ( Ann ℜ ( N ) ) ) , which implies that ν s ( N ) = ν s ( Ann ℑ ( Ann ℜ ( N ) ) ) . Now, the result is clear by (5).

(7) First, suppose that ℑ is a comultiplication module. By Theorem 2.1(5), we have ν s * ( sec ( N ) ) = ν s * ( N ) . So, we obtain

ν s ( N ) = ν s * ( N ) = ν s * ( sec ( N ) ) = ν s ( sec ( N ) ) ,

by (4). For the second case, suppose that N ∈ Spec L ( ℑ ) . Since sec ( N ) ⊆ N , we have ν s ( sec ( N ) ) ⊆ ν s ( N ) . Let K ∈ ν s ( N ) . It is enough to show that Ann ℜ ( sec ( K ) ) ⊆ Ann ℜ ( K ) . Since K ∈ ν s ( N ) , we have Ann ℜ ( N ) ⊆ Ann ℜ ( K ) . This implies that

Ann ℜ ( sec ( N ) ) = Ann ℜ ( N ) ⊆ Ann ℜ ( K ) = Ann ℜ ( K ) ,

as desired.□

Remark that the reverse inclusion in Lemma 2.4(4) is not always true. For example, take Im = Q × Q as Q -module, where Q is the field of rational numbers. Let N = Q × { 0 } and L = { 0 } × Q . By some computations, we can see that L ∈ ν s ( N ) − ν s * ( N ) .

Corollary 2.5

Every secondary cotop module is a cotop module.

Proof

Let ℑ be a secondary cotop module and N 1 , N 2 ≤ ℑ . By Lemma 2.4(2), we have V s * ( N 1 ) ∪ V s * ( N 2 ) = ( Spec s ( ℑ ) ∩ ν s * ( N 1 ) ) ∪ ( Spec s ( ℑ ) ∩ ν s * ( N 2 ) ) = Spec s ( ℑ ) ∩ ( ν s * ( N 1 ) ∪ ν s * ( N 2 ) ) = Spec s ( ℑ ) ∩ ν s * ( T ) = V s * ( T ) for some submodule T of ℑ . Therefore, ℑ is a cotop module.□

Let ℑ be an ℜ -module. By Corollary 2.5, if ℑ is a secondary cotop module, then ζ s * ( ℑ ) = { V s * ( N ) ∣ N ≤ ℑ } generates the quasi-Zariski topology on Spec s ( ℑ ) . By Lemma 2.4(2), Spec s ( ℑ ) equipped with this topology is a topological subspace of Spec L ( ℑ ) equipped with the quasi-Zariski topology. In addition, by Lemma 2.4(1), Spec L ( ℑ ) with the Sℒ -topology contains Spec s ( ℑ ) with the Zariski topology as a topological subspace.

Let ℑ be an ℜ -module. Throughout the rest of this article, we assume that both Spec s ( ℑ ) and Spec L ( ℑ ) are non-empty sets unless stated otherwise and are equipped with the Zariski topology. We also consider ψ and φ as defined in the introduction.

Let ℑ be an ℜ -module and p be a prime ideal of ℜ . We set Spec p L ( ℑ ) = { K ∈ Spec L ( ℑ ) ∣ Ann ℜ ( K ) = p } .

Proposition 2.6

The following statements are equivalent for any ℜ -module ℑ :

  1. If K , K ′ ∈ Spec L ( ℑ ) with ν s ( K ) = ν s ( K ′ ) , then K = K ′ .

  2. ∣ Spec p L ( ℑ ) ∣ ≤ 1 for any p ∈ Spec ( ℜ ) .

  3. φ is injective.

Proof

(1) ⇒ (2): Let p ∈ spec ( ℜ ) and K , K ′ ∈ Spec p L ( ℑ ) . Then, K , K ′ ∈ Spec L ( ℑ ) and Ann ℜ ( K ) = Ann ℜ ( K ′ ) = p . By Lemma 2.4(3), we obtain ν s ( K ) = ν s ( K ′ ) , and thus, K = K ′ by assumption (1).

(2) ⇒ (3): Suppose that φ ( K ) = φ ( K ′ ) , where K , K ′ ∈ Spec L ( ℑ ) . Then, Ann ℜ ( K ) = Ann ℜ ( K ′ ) . Let p = Ann ℜ ( K ) ∈ Spec ( ℜ ) . Then, K , K ′ ∈ Spec p L ( ℑ ) , and by the hypothesis, we obtain K = K ′ .

(3) ⇒ (1): Let K , K ′ ∈ Spec L ( ℑ ) with ν s ( K ) = ν s ( K ′ ) . By Lemma 2.4(3), we have Ann ℜ ( K ) = Ann ℜ ( K ′ ) . So, φ ( K ) = φ ( K ′ ) , and so K = K ′ as φ is injective.□

The following is an easy result for Proposition 2.6.

Corollary 2.7

Let ℑ be an ℜ -module. If  ∣ Spec p L ( ℑ ) ∣ = 1 for any p ∈ Spec ( ℜ ) , then φ is bijective.

Let ℑ be an ℜ -module. From now on, for any ideal I of ℜ containing Ann ℜ ( ℑ ) , I ¯ and ℜ ¯ will denote I ∕ Ann ℜ ( ℑ ) and ℜ ∕ Ann ℜ ( ℑ ) , respectively. In addition, the class r + Ann ℜ ( ℑ ) in ℜ ∕ Ann ℜ ( ℑ ) will be expressed by r ¯ for any r ∈ ℜ .

In the following lemma, we list some properties of the natural map ψ of Spec L ( ℑ ) , which are needed in the rest of this article.

Lemma 2.8

([11, Proposition 3.6 and Theorem 3.11]) The following hold for any ℜ -module ℑ :

  1. The natural map ψ of Spec s ( ℑ ) is continuous and ψ − 1 ( V ℜ ¯ ( I ¯ ) ) = V s ( Ann ℑ ( I ) ) for any ideal I of ℜ containing Ann ℜ ( ℑ ) .

  2. If ψ is surjective, then ψ is both open and closed with ψ ( V s ( N ) ) = V ℜ ¯ ( Ann ℜ ( N ) ¯ ) and ψ ( Spec s ( ℑ ) − V s ( N ) ) = Spec ( ℜ ¯ ) − V ℜ ¯ ( Ann ℜ ( N ) ) for any submodule N of ℑ .

In the next two propositions, we provide similar results for φ .

Proposition 2.9

Let ℑ be an ℜ -module. Then, φ − 1 ( V ℜ ¯ ( I ¯ ) ) = ν s ( Ann ℑ ( I ) ) for any ideal I of ℜ containing Ann ℜ ( ℑ ) . Therefore, φ is continuous.

Proof

For any K ∈ Spec L ( ℑ ) , we have K ∈ φ − 1 ( V ℜ ¯ ( I ¯ ) ) ⇔ φ ( K ) = Ann ℜ ( K ) ¯ ∈ V ℜ ¯ ( I ¯ ) ⇔ I ¯ ⊆ Ann ℜ ( K ) ¯ ⇔ I ⊆ Ann ℜ ( K ) = Ann ℜ ( sec ( K ) ) ⇔ sec ( K ) ⊆ Ann ℑ ( I ) ⇔ Ann ℜ ( Ann ℑ ( I ) ) ⊆ Ann ℜ ( sec ( K ) )   ⇔ K ∈ ν s ( Ann ℑ ( I ) ) . Therefore, φ − 1 ( V ℜ ¯ ) = ν s ( Ann ℑ ( I ) ) .□

Proposition 2.10

Let ℑ be an ℜ -module. If φ is surjective, then φ is both closed and open. In particular, φ ( ν s ( N ) ) = V ℜ ¯ ( Ann ℜ ( N ) ¯ ) and φ ( Spec L ( ℑ ) − ν s ( N ) ) = Spec ( ℜ ¯ ) − V ℜ ¯ ( Ann ℜ ( N ) ¯ ) for any submodule N of ℑ .

Proof

Using Proposition 2.9, we obtain φ − 1 ( V ℜ ¯ ( I ¯ ) ) = ν s ( Ann ℑ ( I ) ) for any ideal I of ℜ containing Ann ℜ ( ℑ ) , which implies that φ − 1 ( V ℜ ¯ ( Ann ℜ ( N ) ¯ ) ) = ν s ( Ann ℑ ( Ann ℜ ( N ) ) ) = ν s ( N ) by Lemma 2.4(6). Since φ is surjective, we have φ ( ν s ( N ) ) = V ℜ ¯ ( Ann ℜ ( N ) ¯ ) . Hence, φ is closed. To show that φ is an open map, note that Spec L ( ℑ ) − ν s ( N ) = Spec L ( ℑ ) − φ − 1 ( V ℜ ¯ ( Ann ℜ ( N ) ¯ ) ) = φ − 1 ( Spec ( ℜ ¯ ) − V ℜ ¯ ( Ann ℜ ( N ) ¯ ) ) . Again, by the surjectivity of φ , we obtain φ ( Spec L ( ℑ ) − ν s ( N ) ) = Spec ( ℜ ¯ ) − V ℜ ¯ ( Ann ℜ ( N ) ¯ ) , as desired.□

The proof of the following result is straightforward using Propositions 2.9 and 2.10.

Corollary 2.11

Let ℑ be an ℜ -module. Then, φ is bijective if and only if φ is a homeomorphism.

Recall that a topological space X is connected if the only clopen subsets of X are X and the empty set. In the following theorem, we relate the connectedness of Spec s ( ℑ ) to the connectedness of both Spec ( ℜ ¯ ) and Spec s ( ℑ ) for any ℜ -module ℑ using the properties of φ and ψ that are stated in Lemma 2.8 and Propositions 2.9 and 2.10.

Theorem 2.12

Let ℑ be an ℜ -module, and consider the following statements:

  1. Spec s ( ℑ ) is connected.

  2. Spec L ( ℑ ) is connected.

  3. Spec ( ℜ ¯ ) is connected.

  4. 0 ¯ and 1 ¯ are the only idempotent elements of ℜ ¯ .

  1. If φ is surjective, then (1) ⇒ (2) ⇔ (3) ⇔ (4).

  2. If ψ is surjective, then all the four statements are equivalent.

Proof

(i) (1) ⇒ (2): Suppose that Spec s ( ℑ ) is connected. If Spec L ( ℑ ) is disconnected, then there exists W clopen in Spec L ( ℑ ) such that W ≠ Spec L ( ℑ ) and W ≠ ∅ . Since W is clopen in Spec L ( ℑ ) , we have W = Spec L ( ℑ ) − ν s ( N ) = ν s ( N ′ ) for some submodules N and N ′ of ℑ . By Proposition 2.10, φ ( W ) is clopen in Spec ( ℜ ¯ ) . But by Lemma 2.8(1), ψ is continuous. So ψ − 1 ( φ ( W ) ) is clopen in Spec s ( ℑ ) , and thus, ψ − 1 ( φ ( W ) ) = ∅ or ψ − 1 ( φ ( W ) ) = Spec s ( ℑ ) . If ψ − 1 ( φ ( W ) ) = ∅ , then ψ − 1 ( φ ( ν s ( N ′ ) ) ) = ∅ , and hence, ψ − 1 ( V ℜ ¯ ( Ann ℜ ( N ′ ) ¯ ) ) = ∅ , which implies that

V s ( N ′ ) = V s ( Ann ℑ ( Ann ℜ ( N ′ ) ) ) = ∅ ,

using [12, Lemma 3.3(b)] and Lemma 2.8(1). This implies that Ann ℜ ( N ′ ) ⊈ Ann ℜ ( S ) for any S ∈ Spec s ( ℑ ) . Since W = ν s ( N ′ ) ≠ ∅ , there exists K ∈ Spec L ( ℑ ) such that Ann ℜ ( N ′ ) ⊆ Ann ℜ ( K ) = Ann ℜ ( sec ( K ) ) . As sec ( K ) ≠ { 0 } , then there exists S ′ ∈ Spec s ( ℑ ) such that S ′ ⊆ K . Hence, Ann ℜ ( K ) ⊆ Ann ℜ ( S ′ ) = Ann ℜ ( S ′ ) ∈ Spec ( ℜ ) , and thus, Ann ℜ ( N ′ ) ⊆ Ann ℜ ( S ′ ) , which is a contradiction. Now, if ψ − 1 ( φ ( W ) ) = Spec s ( ℑ ) , then Spec s ( ℑ ) = ψ − 1 ( φ ( ν s ( N ′ ) ) ) = ψ − 1 ( V ℜ ¯ ( Ann ℜ ( N ′ ) ¯ ) ) = V s ( Ann ℑ ( Ann ℜ ( N ′ ) ) ) = V s ( N ′ ) by Lemma 2.8(1) and [12, Lemma 3.3(b)], again. But by Lemma 2.4(1), we have V s ( N ′ ) = Spec s ( ℑ ) ∩ ν s ( N ′ ) , which implies that Spec s ( ℑ ) ⊆ ν s ( N ′ ) = W = Spec L ( ℑ ) − ν s ( N ) . Since W = Spec L ( ℑ ) − ν s ( N ) ≠ Spec L ( ℑ ) , there exists K ∈ Spec L ( ℑ ) such that Ann ℜ ( N ) ⊆ Ann ℜ ( K ) . As Soc ( K ) ≠ { 0 } , then there exists S ∈ Spec s ( ℑ ) such that S ⊆ K . So we have Ann ℜ ( N ) ⊆ Ann ℜ ( K ) ⊆ Ann ℜ ( S ) and so S ∈ ν s ( N ) . But S ∈ Spec s ( ℑ ) ⊆ Spec L ( ℑ ) − ν s ( N ) . Therefore, S ∉ ν s ( N ) , which is a contradiction. Hence, Spec L ( ℑ ) is connected.

(2) ⇒ (3): This follows from the fact that the continuous image of a connected space is also connected.

(3) ⇒ (2): Suppose the contrary, i.e., Spec L ( ℑ ) is disconnected. Then, there exists a clopen set W in Spec L ( ℑ ) such that W ≠ ∅ and W ≠ Spec L ( ℑ ) . By Proposition 2.10, φ ( W ) is clopen in Spec ( ℜ ¯ ) , which is a connected space by assumption (3). This implies that φ ( W ) = Spec ( ℜ ¯ ) or φ ( W ) = ∅ . Since W is open in Spec L ( ℑ ) , we have W = Spec L ( ℑ ) − ν s ( N ) for some N ≤ ℑ . Note that if φ ( W ) = Spec ( ℜ ¯ ) , then Spec ( ℜ ¯ ) = φ ( Spec L ( ℑ ) − ν s ( N ) ) = Spec ( ℜ ¯ ) − V ℜ ¯ ( Ann ℜ ( N ) ¯ ) . Thus V ℜ ¯ ( Ann ℜ ( N ) ¯ ) = ∅ , and hence, ∅ = φ − 1 ( ∅ ) = φ − 1 ( V ℜ ¯ ( Ann ℜ ( N ) ¯ ) ) = ν s ( N ) by Proposition 2.9. This implies that ν s ( N ) = ∅ , and hence, W = Spec L ( ℑ ) , which is a contradiction. Now, if φ ( W ) = ∅ , then W ⊆ φ − 1 ( φ ( W ) ) = ∅ , and thus, W = ∅ , which is a contradiction. Consequently, Spec L ( ℑ ) is connected.

The equivalence of (3) and (4) follows from [2, p. 132, Corollary 2 to Proposition 15].

(ii) It is easy to see that if ψ is surjective, then φ is surjective, and thus, (1) ⇒ (2) ⇔ (3) ⇔ (4) by part (i). To complete the proof, it is enough to show that (4) ⇒ (1). But this follows from [11, Corollary 3.13].□

Let f : ℑ → ℑ ′ be a module monomorphism of ℜ -modules. Let N ≤ ℑ and N ′ ≤ ℑ ′ be such that N ′ ⊆ f ( ℑ ) . By [8, Lemma 2.19], we have f ( sec ( N ) ) = sec ( f ( N ) ) and f − 1 ( sec ( N ′ ) ) = sec ( f − 1 ( N ′ ) ) . In addition, it is easy to see that Ann ℜ ( N ) = Ann ℜ ( f ( N ) ) and Ann ℜ ( N ′ ) = Ann ℜ ( f − 1 ( N ′ ) ) . Now, we use these assertions to prove the next two results.

Lemma 2.13

Let f : ℑ → ℑ ′ be a module monomorphism of ℜ -modules. Then, the following hold:

  1. If N ′ ∈ Spec L ( ℑ ′ ) is such that N ′ ⊆ f ( ℑ ) , then f − 1 ( N ′ ) ∈ Spec L ( ℑ ) .

  2. If N ∈ Spec L ( ℑ ) , then f ( N ) ∈ Spec L ( ℑ ′ ) .

Proof

(1) Since N ′ ≠ { 0 ℑ ′ } and N ′ ⊆ f ( ℑ ) , we have f − 1 ( N ′ ) ≠ { 0 ℑ } . Now, for any r ∈ ℜ , we have r N ′ = N ′ or r ∈ Ann ℜ ( N ′ ) as N ′ is a secondary submodule of ℑ ′ . Thus, r f − 1 ( N ′ ) = f − 1 ( r N ′ ) = f − 1 ( N ′ ) or r ∈ Ann ℜ ( N ′ ) = Ann ℜ ( f − 1 ( N ′ ) ) . This implies that f − 1 ( N ′ ) is a secondary submodule of ℑ . Since Ann ℜ ( sec ( N ′ ) ) = Ann ℜ ( N ′ ) and N ′ ⊆ f ( ℑ ) , we have Ann ℜ ( sec ( f − 1 ( N ′ ) ) ) = Ann ℜ ( f − 1 ( sec ( N ′ ) ) ) = Ann ℜ ( sec ( N ′ ) ) = Ann ℜ ( N ′ ) = Ann ℜ ( f − 1 ( N ′ ) ) . This means that f − 1 ( N ′ ) ∈ Spec L ( ℑ ) , as desired.

(2) It is easy to check that f ( N ) is a secondary submodule of ℑ ′ . Now, since N ∈ Spec L ( ℑ ) , we have Ann ℜ ( sec ( N ) ) = Ann ℜ ( N ) , which implies that Ann ℜ ( sec ( f ( N ) ) ) = Ann ℜ ( f ( sec ( N ) ) ) = Ann ℜ ( sec ( N ) ) = Ann ℜ ( N ) = Ann ℜ ( f ( N ) ) . Therefore, f ( N ) ∈ Spec L ( ℑ ′ ) , as needed.□

Theorem 2.14

Let f : ℑ → ℑ ′ be a module monomorphism of ℜ -modules. Then, the map ρ : Spec L ( ℑ ) → Spec L ( ℑ ′ ) defined by ρ ( K ) = f ( K ) is an injective continuous map. Moreover, if ρ is surjective, then Spec L ( ℑ ) is homeomorphic to Spec L ( ℑ ′ ) .

Proof

By Lemma 2.13(2), ρ is well defined. In addition, the injectivity of f implies the injectivity of ρ . By Lemma 2.4(5) and (6), we obtain that for any K ∈ Spec L ( ℑ ) and any closed set ν s ( N ′ ) in Spec L ( ℑ ′ ) , where N ′ ≤ ℑ ′ , we have K ∈ ρ − 1 ( ν s ( N ′ ) ) = ρ − 1 ( ν s * ( Ann ℑ ′ ( Ann ℜ ( N ′ ) ) ) ) ⇔ sec ( f ( K ) ) ⊆ Ann ℑ ′ ( Ann ℜ ( N ′ ) ) ⇔ Ann ℜ ( N ′ ) ⊆ Ann ℜ ( sec ( f ( K ) ) ) = Ann ℜ ( f ( sec ( K ) ) ) = Ann ℜ ( sec ( K ) ) ⇔ sec ( K ) ⊆ Ann ℑ ( Ann ℜ ( N ′ ) ) ⇔ K ∈ ν s * ( Ann ℑ ( Ann ℜ ( N ′ ) ) ) = ν s ( Ann ℑ ( Ann ℜ ( N ′ ) ) ) . Thus,

ρ − 1 ( ν s ( N ′ ) ) = ν s ( Ann ℑ ( Ann ℜ ( N ′ ) ) ) ,

and hence, ρ is continuous. Now, suppose that ρ is surjective. To complete the proof, it is sufficient to show that ρ is closed. So, let ν s ( N ) be any closed set in Spec L ( ℑ ) , where N ≤ ℑ . As we have seen ρ − 1 ( ν s ( N ′ ) ) = ν s ( Ann ℑ ( Ann ℜ ( N ′ ) ) ) for any submodule N ′ of ℑ ′ . So, ρ − 1 ( ν s ( f ( N ) ) ) = ν s ( Ann ℑ ( Ann ℜ ( f ( N ) ) ) ) = ν s ( Ann ℑ ( Ann ℜ ( N ) ) ) = ν s ( N ) by Lemma 2.4(6). Since ρ is surjective, we have ρ ( ν s ( N ) ) = ν s ( f ( N ) ) . This implies that ρ is closed. Consequently, Spec L ( ℑ ) is homeomorphic to Spec L ( ℑ ′ ) .□

The proof of the following result is straightforward by using Lemma 2.13(1) and Theorem 2.14.

Corollary 2.15

Let f : ℑ → ℑ ′ be a module isomorphism of ℜ -modules. Then, Spec L ( ℑ ) is homeomorphic to Spec L ( ℑ ′ ) .

3 Base for the Zariski topology on Spec L ( ℑ )

Let ℑ be an ℜ -module, and let D r = Spec ( ℜ ) − V ℜ ( r R ) for r ∈ ℜ . It is known that the collection { D r ∣ r ∈ ℜ } is a base for the Zariski topology on Spec ( ℜ ) . In addition, each D r is quasi-compact, and thus, Spec ( ℜ ) = D 1 is quasi-compact.

In this section, we put E r = Spec L ( ℑ ) − ν s ( Ann ℑ ( r ) ) for each r ∈ ℜ and show that E = { E r ∣ r ∈ ℜ } forms a base for the Sℒ -topology and give some related results. It is clear that E 0 = ∅ and E 1 = Spec L ( ℑ ) .

Theorem 3.1

Let ℑ be an ℜ -module. Then, the set E = { E r ∣ r ∈ ℜ } forms a base for Spec L ( ℑ ) with the Sℒ -topology.

Proof

Let W = Spec L ( ℑ ) − ν s ( N ) be an open set in Spec L ( ℑ ) , where N ≤ ℑ and let K ∈ W . To complete the proof, and it is sufficient to find r ∈ ℜ such that K ∈ E r ⊆ W . As K ∈ W , then Ann ℜ ( N ) ⊈ Ann ℜ ( K ) = Ann ℜ ( sec ( K ) ) , and hence, there exists r ∈ Ann ℜ ( N ) − Ann ℜ ( sec ( K ) ) . If K ∉ E r , then r R ⊆ Ann ℜ ( Ann ℑ ( r ) ) ⊆ Ann ℜ ( sec ( K ) ) , and hence, r ∈ Ann ℜ ( sec ( K ) ) , which is a contradiction, and this means that K ∈ E r . As r ∈ Ann ℜ ( N ) , then Ann ℑ ( Ann ℜ ( N ) ) ⊆ Ann ℑ ( r ) . Now, we prove that E r ⊆ W . So let K ′ ∈ E r . Then, Ann ℜ ( Ann ℑ ( r ) ) ⊈ Ann ℜ ( sec ( K ′ ) ) . If K ′ ∉ W , then Ann ℜ ( N ) ⊆ Ann ℜ ( sec ( K ′ ) ) , which implies that Ann ℑ ( Ann ℜ ( sec ( K ′ ) ) ) ⊆ Ann ℑ ( Ann ℜ ( N ) ) ⊆ Ann ℑ ( r ) . So

Ann ℜ ( Ann ℑ ( r ) ) ⊆ Ann ℜ ( Ann ℑ ( Ann ℜ ( sec ( K ′ ) ) ) ) = Ann ℜ ( sec ( K ′ ) ) ,

which is a contradiction. Therefore, K ′ ∈ W . Consequently, K ∈ E r ⊆ W , as needed.□

For any ring ℜ , the set of all units in ℜ and the nilradical of ℜ will be denoted by U ( ℜ ) and N ( ℜ ) , respectively.

Proposition 3.2

Let ℑ be an ℜ -module and r ∈ ℜ . Then, we have the following:

  1. φ − 1 ( D r ¯ ) = E r .

  2. φ ( E r ) ⊆ D r ¯ . If φ is surjective, then the equality holds.

  3. E a ∩ E b = E a b for any elements a , b ∈ ℜ .

  4. If r ∈ N ( ℜ ) , then E r = ∅ .

  5. If r ∈ U ( ℜ ) , then E r = Spec L ( ℑ ) .

Proof

  1. φ − 1 ( D r ¯ ) = φ − 1 ( Spec ( ℜ ¯ ) − V ℜ ¯ ( r ¯ ℜ ¯ ) ) = Spec L ( ℑ ) − φ − 1 ( V ℜ ¯ ( r ¯ ℜ ¯ ) ) = Spec L ( ℑ ) − ν s ( Ann ℑ ( r ) ) = E r by Proposition 2.9.

  2. follows from (1).

  3. E a ∩ E b = φ − 1 ( D a ¯ ) ∩ φ − 1 ( D b ¯ ) = φ − 1 ( D a ¯ ∩ D b ¯ ) = φ − 1 ( D a b ¯ ) = E a b .

  4. Suppose that r ∈ N ( ℜ ) . Then, D r = ∅ , and hence, D r ¯ = ∅ . Thus, E r = φ − 1 ( D r ¯ ) = ∅ .

  5. Suppose that r ∈ U ( ℜ ) . Then, D r = Spec ( ℜ ) , which implies that D r ¯ = Spec ( ℜ ¯ ) . Therefore, E r = φ − 1 ( D r ¯ ) = φ − 1 ( Spec ( ℜ ¯ ) ) = Spec L ( ℑ ) .□

Corollary 3.3

Let ℑ be an ℜ -module. If ℜ is field, then the Zariski topology on Spec L ( ℑ ) is the trivial topology.

Proof

For any r ∈ ℜ − { 0 } , we have r ∈ U ( ℜ ) , and hence, E r = Spec L ( ℑ ) by Proposition 3.2(5). Also, E 0 = ∅ . So E = { E r ∣ r ∈ ℜ } = { Spec L ( ℑ ) , ∅ } , and so the Zariski topology on Spec L ( ℑ ) is the trivial topology.□

The converse of the aforementioned result is not true in general. For example, take ℑ 2 ( Z 8 ) (the group of all 2 × 2 metrices with entries in Z 8 ) as Z 8 -module. Then, 1 , 3 , 5 , 7 ∈ U ( Z 8 ) and 0 , 2 , 4 , 6 ∈ N ( Z 8 ) , which implies that E 1 = E 3 = E 5 = E 7 = Spec L ( ℑ 2 ( Z 8 ) ) and E 0 = E 2 = E 4 = E 6 = ∅ by Proposition 3.2. So, E = { E r ∣ r ∈ ℜ } = { ∅ , Spec L ( ℑ 2 ( Z 8 ) ) } . Therefore, the Zariski topology on Spec L ( ℑ 2 ( Z 8 ) ) is the trivial topology. However, Z 8 is not field.

Theorem 3.4

Let ℑ be an ℜ -module. If φ is surjective, then for each r ∈ ℜ , E r is quasi-compact. Therefore, Spec L ( ℑ ) is quasi-compact.

Proof

Let r ∈ ℜ and ℬ = { E a ∣ a ∈ Λ } be a basic open cover for E r , where Λ ⊆ ℜ . This implies that E r ⊆ ⋃ a ∈ Λ E a , and thus, D r ¯ = φ ( E r ) ⊆ ⋃ a ∈ Λ φ ( E a ) = ⋃ a ∈ Λ D a ¯ using Proposition 3.2(2). So ℬ ¯ = { D a ¯ ∣ a ∈ Λ } is a basic open cover for D r ¯ , which is a quasi-compact set in Spec ( ℜ ) , and so, it has a finite subcover ℬ * = { D a j ¯ ∣ j = 1 , … , m } , where m ∈ N and t j ∈ Λ for any j = 1 , … , m . Thus, D r ¯ ⊆ ⋃ j = 1 m D a j ¯ , which implies that E r = φ − 1 ( D r ¯ ) ⊆ ⋃ j = 1 m φ − 1 ( D a j ¯ ) = ⋃ j = 1 m E a j ¯ using Proposition 3.2(1). Therefore, ℬ * * = { E a j ¯ ∣ j = 1 , … , m } ⊆ ℬ is a finite subcover for E r . The last part of the theorem follows from the equality Spec L ( ℑ ) = E 1 .□

Theorem 3.5

Let ℑ be an ℜ -module, and let φ be surjective. Then, the family of all quasi-compact open subsets of Spec L ( ℑ ) is closed under finite intersections, and it is an open base of Spec L ( ℑ ) with the Sℒ -topology.

Proof

Let C 1 and C 2 be quasi-compact open subsets of Spec L ( ℑ ) with the Sℒ -topology. Let i ∈ { 1 , 2 } . Since, by Theorem 3.1, the family E is an open base for Spec L ( ℑ ) , the set C i is the union of some subfamily of E . Thus, it follows from the quasi-compactness of C i that C i is the union of a finite subfamily of E . Hence, in view of Proposition 3.2(3), the set C 1 ∩ C 2 is the union of a finite subfamily of E . By Theorem 3.4, each member of E is quasi-compact, so C 1 ∩ C 2 is a finite union of quasi-compact sets. It is well known that finite unions of quasi-compact sets are quasi-compact. In consequence, C 1 ∩ C 2 is quasi-compact.□

4 Irreducibility in Spec L ( ℑ )

Let ℑ be an ℜ -module and Y ⊆ Spec L ( ℑ ) . The closure of Y in Spec L ( ℑ ) will be denoted by Cl ( Y ) . We also denote the sum ∑ K ∈ Y sec ( K ) by H ( Y ) . If Y = ∅ , we set H ( Y ) = { 0 } .

Proposition 4.1

For any ℜ -module ℑ and Y ⊆ Spec L ( ℑ ) , we have Cl ( Y ) = ν s ( H ( Y ) ) . Therefore, Y is closed in Spec L ( ℑ ) ⇔ ν s ( H ( Y ) ) = Y . Moreover, if ℑ ∈ Y , then Y is dense in Spec L ( ℑ ) .

Proof

Clearly, Y ⊆ ν s ( H ( Y ) ) . Let ν s ( N ) be any closed set containing Y , and it is sufficient to show that ν s ( H ( Y ) ) ⊆ ν s ( N ) . So let K ∈ ν s ( H ( Y ) ) . Then, Ann ℜ ( H ( Y ) ) ⊆ Ann ℜ ( sec ( K ) ) . But for any K ′ ∈ Y , we have Ann ℜ ( N ) ⊆ Ann ℜ ( sec ( K ′ ) ) , and hence, Ann ℜ ( N ) ⊆ ⋂ Q ∈ Y Ann ℜ ( sec ( Q ) ) = Ann ℜ ( ∑ Q ∈ Y sec ( Q ) ) = Ann ℜ ( H ( Y ) ) ⊆ Ann ℜ ( sec ( K ) ) . Thus, K ∈ ν s ( N ) . This means that ν s ( H ( Y ) ) is the smallest closed set containing Y , and hence, Cl ( Y ) = ν s ( H ( Y ) ) . For the last statement, if ℑ ∈ Y , then Cl ( Y ) = ν s ( H ( Y ) ) = ν s ( sec ( ℑ ) ) = ν s ( ℑ ) = Spec L ( ℑ ) using Lemma 2.4(7).□

Recall that a topological space Z is said to be irreducible if Z ≠ ∅ and whenever Z = Z 1 ∪ Z 2 with closed sets Z 1 and Z 2 in Z , then either Z = Z 1 or Z = Z 2 . Let Z ′ ⊆ Z . Then, Z ′ is irreducible if it is irreducible as a subspace of Z . The maximal irreducible subsets of Z are called the irreducible components of Z . It is not difficult to prove that any singleton subset of Z is irreducible. Also, a subset W of Z is irreducible if and only if Cl ( W ) is irreducible (see [2,16]).

Theorem 4.2

Let ℑ be an ℜ -module, and let K ∈ Spec L ( ℑ ) . Then, Cl ( { K } ) = ν s ( K ) and ν s ( K ) is an irreducible closed subset of Spec L ( ℑ ) . In particular, if ℑ ∈ Spec L ( ℑ ) , then Spec L ( ℑ ) is irreducible.

Proof

By Lemma 2.4(7) and Proposition 4.1, Cl ( { K } ) = ν s ( H ( K ) ) = ν s ( sec ( K ) ) = ν s ( K ) . Now, since { K } is irreducible, we have Cl ( { K } ) = ν s ( K ) is irreducible. The last statement follows from the equality Spec L ( ℑ ) = ν s ( ℑ ) .□

Note that ν s ( N ) for a submodule N of an ℜ -module, ℑ is not always irreducible. In fact, Spec L ( ℑ ) might not be irreducible. For example, take Z 6 as Z 6 -module. By some computations, we can see that Spec L ( Z 6 ) = { { 0 , 2 , 4 } , { 0 , 3 } } , ν s ( 3 Z 6 ) = { { 0 , 3 } } and ν s ( 2 Z 6 ) = { { 0 , 2 , 4 } } . Note that Spec L ( Z 6 ) = ν s ( 2 Z 6 ) ∪ ν s ( 3 Z 6 ) . But Spec L ( Z 6 ) ≠ ν s ( 2 Z 6 ) and Spec L ( Z 6 ) ≠ ν s ( 3 Z 6 ) . Hence, Spec L ( Z 6 ) = ν s ( Z 6 ) is not irreducible.

Corollary 4.3

Let ℑ be an ℜ -module and Y ⊆ Spec L ( ℑ ) such that Ann ℜ ( H ( Y ) ) = p is a prime ideal of ℜ . If Spec p L ( ℑ ) ≠ ∅ , then Y is irreducible in Spec L ( ℑ ) .

Proof

Let K ∈ Spec p L ( ℑ ) . Then, Ann ℜ ( K ) = p = Ann ℜ ( H ( Y ) ) , and thus, ν s ( K ) = ν s ( H ( Y ) ) = Cl ( Y ) by Lemma 2.4(3) and Proposition 4.1. Using Theorem 4.2, Cl ( Y ) = ν s ( K ) is irreducible, and hence, Y is irreducible.□

Let ℜ be a ring and Y ⊆ Spec ( ℜ ) . If Y ≠ ∅ , the intersection of all members of Y is denoted by ξ ( Y ) . If Y = ∅ , we set ξ ( Y ) = ∅ .

Theorem 4.4

Let ℑ be an ℜ -module and ∅ ≠ Y ⊆ Spec L ( ℑ ) . Then:

  1. If H ( Y ) is a secondary submodule of ℑ , then Y is irreducible.

  2. If Y is irreducible, then ϒ = { Ann ℜ ( sec ( K ) ) ∣ K ∈ Y } is an irreducible closed subset of Spec ( ℜ ) , i.e., ξ ( ϒ ) = Ann ℜ ( H ( Y ) ) is a prime ideal of ℜ .

Proof

(1) Suppose that H ( Y ) is a secondary submodule of ℑ . By [3, Proposition 2.1(h)], we have sec ( H ( Y ) ) ⊆ Ann ℑ ( Ann ℜ ( H ( Y ) ) ) , and hence, Ann ℜ ( H ( Y ) ) ⊆ Ann ℜ ( Ann ℑ ( Ann ℜ ( H ( Y ) ) ) ) ⊆ Ann ℜ ( sec ( H ( Y ) ) ) . Thus, Ann ℜ ( H ( Y ) ) ⊆ Ann ℜ ( sec ( H ( Y ) ) ) . But sec ( H ( Y ) ) = H ( Y ) . This implies that Ann ℜ ( sec ( H ( Y ) ) ) = Ann ℜ ( H ( Y ) ) ⊆ Ann ℜ ( H ( Y ) ) . This means that Ann ℜ ( sec ( H ( Y ) ) ) = Ann ℜ ( H ( Y ) ) , and hence, H ( Y ) ∈ Spec L ( ℑ ) . By Theorem 4.2, Cl ( Y ) = ν s ( H ( Y ) ) is irreducible in Spec L ( ℑ ) , and thus, Y is irreducible, as desired.

(2) Suppose that Y is irreducible in Spec L ( ℑ ) . So φ ( Y ) = Y ′ is an irreducible subset of Spec ( ℜ ¯ ) as φ is continuous using Proposition 2.9. Now,

ξ ( Y ′ ) = ξ ( φ ( Y ) ) = ⋂ K ∈ Y Ann ℜ ( sec ( K ) ) ¯ = ⋂ K ∈ Y Ann ℜ ( sec ( K ) ) ¯ = Ann ℜ ∑ K ∈ Y sec ( K ) ¯ = Ann ℜ ( H ( Y ) ) ¯ .

Since Y ′ is irreducible in Spec ( ℜ ¯ ) , we have ξ ( Y ′ ) = Ann ℜ ( H ( Y ) ) ¯ ∈ Spec ( ℜ ¯ ) by [2, p. 129, Proposition 14]. This implies that

ξ ( ϒ ) = ⋂ K ∈ Y Ann ℜ ( sec ( K ) ) = Ann ℜ ( H ( Y ) ) ∈ Spec ( ℜ ) .

Again, ϒ is an irreducible subset of Spec ( ℜ ) by [2, p. 129, Proposition 14].□

Let Z be a topological space and F be a closed subset of Z . Recall that an element a ∈ F is called a generic point of F if Cl ( { a } ) = F .

Theorem 4.5

Let ℑ be an ℜ -module and Y ⊆ Spec L ( ℑ ) . If φ is surjective, then Y is an irreducible closed subset of Spec L ( ℑ ) if and only if Y = ν s ( K ) for some K ∈ Spec L ( ℑ ) . Therefore, every non-empty irreducible closed subset of Spec L ( ℑ ) has a generic point.

Proof

Suppose that ∅ ≠ Y is an irreducible closed subset of Spec L ( ℑ ) . Then, Ann ℜ ( H ( Y ) ) ∈ Spec ( ℜ ) by Theorem 4.4(2), and hence, Ann ℜ ( H ) ¯ ∈ Spec ( ℜ ¯ ) . As φ is surjective, we obtain Ann ℜ ( H ( Y ) ) = Ann ℜ ( sec ( K ) ) for some K ∈ Spec L ( ℑ ) . This implies that Ann ℜ ( K ) = Ann ℜ ( H ( Y ) ) , and thus, ν s ( K ) = ν s ( H ( Y ) ) = Cl ( Y ) = Y by Lemma 2.4(3) and Proposition 4.1. Conversely, if Y = ν s ( K ) for some K ∈ Spec L ( ℑ ) , then Y is closed and it is irreducible using Theorem 4.2.□

Theorem 4.6

Let ℑ be an ℜ -module and K ∈ Spec L ( ℑ ) . If Ann ℜ ( sec ( K ) ) ¯ is a minimal prime ideal of ℜ ¯ , then ν s ( K ) is an irreducible component of Spec L ( ℑ ) . If φ is surjective, then the converse is also true.

Proof

Using Theorem 4.2, we infer that ν s ( K ) is irreducible. It is sufficient to prove that ν s ( K ) is a maximal irreducible set. So let Y be an irreducible subset of Spec L ( ℑ ) with ν s ( K ) ⊆ Y . It remains to show that Y = ν s ( K ) . As K ∈ ν s ( K ) ⊆ Y , then K ∈ Y , and hence, sec ( K ) ⊆ H ( Y ) , which implies that Ann ℜ ( H ( Y ) ) ¯ ⊆ Ann ℜ ( sec ( K ) ) ¯ . Using Theorem 4.4(2), we obtain Ann ℜ ( H ( Y ) ) ¯ ∈ Spec ( ℜ ¯ ) . But Ann ℜ ( sec ( K ) ) ¯ is a minimal prime ideal of ℜ ¯ . This implies that Ann ℜ ( H ( Y ) ) ¯ = Ann ℜ ( sec ( K ) ) ¯ , and hence, V ℜ ¯ ( Ann ℜ ( H ( Y ) ) ¯ ) = V ℜ ¯ ( Ann ℜ ( sec ( K ) ) ¯ ) . By Lemma 2.4(6), (7), and Proposition 2.9, we obtain ν s ( H ( Y ) ) = φ − 1 ( V ℜ ¯ ( Ann ℜ ( H ( Y ) ) ¯ ) ) = φ − 1 ( V ℜ ¯ ( Ann ℜ ( sec ( K ) ) ¯ ) ) = ν s ( sec ( K ) ) = ν s ( K ) . Since Y ⊆ ν s ( H ( Y ) ) , we have Y ⊆ ν s ( K ) , and thus, Y = ν s ( K ) . For the converse, suppose that φ is surjective. Let p ¯ ∈ Spec ( ℜ ¯ ) such that p ¯ ⊆ Ann ℜ ( sec ( K ) ) ¯ , and it is sufficient to show that p ¯ = Ann ℜ ( sec ( K ) ) ¯ . Since φ is surjective, there exists K ′ ∈ Spec L ( ℑ ) such that Ann ℜ ( sec ( K ′ ) ) = p . Then, Ann ℜ ( sec ( K ′ ) ) ⊆ Ann ℜ ( sec ( K ) ) , and hence, ν s ( K ) ⊆ ν s ( K ′ ) . Since ν s ( K ) is an irreducible component of Spec L ( ℑ ) and ν s ( K ′ ) is irreducible by Theorem 4.2, we have ν s ( K ) = ν s ( K ′ ) . By Lemma 2.4(3), we obtain Ann ℜ ( K ) = Ann ℜ ( K ′ ) , and thus, Ann ℜ ( sec ( K ) ) = Ann ℜ ( sec ( K ′ ) ) . Therefore, p ¯ = Ann ℜ ( sec ( K ′ ) ) ¯ = Ann ℜ ( sec ( K ) ) ¯ , as desired.□

Corollary 4.7

Let ℑ be an ℜ -module. If φ is surjective, then the correspondence ν s ( K ) → Ann ℜ ( sec ( K ) ) ¯ provides a bijection from the set of irreducible components of Spec L ( ℑ ) to the set of minimal prime ideals of ℜ ¯ .

Proof

The proof is straightforward by Theorems 4.5 and 4.6.□

Corollary 4.8

Let ℑ be an ℜ -module such that Spec L ( ℑ ) ≠ ∅ , and let A = { K ∈ Spec L ( ℑ ) ∣ Ann ℜ ( sec ( K ) ) ¯ are a minimal prime ideal of ℜ ¯ } . If φ is surjective, then we have the following:

  1. A ≠ ∅ .

  2. ℬ = { ν s ( K ) ∣ K ∈ A } is the set of all irreducible components of Spec L ( ℑ ) .

  3. Spec L ( ℑ ) = ⋃ K ∈ A ν s ( K ) .

  4. Spec ( ℜ ¯ ) = ⋃ K ∈ A V ℜ ¯ ( Ann ℜ ( K ) ¯ ) .

  5. Spec s ( ℑ ) = ⋃ K ∈ A V s ( K ) .

  6. If ℑ ∈ Spec L ( ℑ ) , then the only irreducible component of Spec L ( ℑ ) is Spec L ( ℑ ) itself.

Proof

  1. Let K ∈ Spec L ( ℑ ) . Since any irreducible subset of a topological space is contained in a maximal irreducible subset of it, there exists an irreducible component Y in Spec L ( ℑ ) such that { K } ⊆ Y . But every irreducible component of a topological space is closed, and hence, Y = ν s ( K ′ ) for some K ′ ∈ Spec L ( ℑ ) by Theorem 4.5. This implies that Ann ℜ ( sec ( K ′ ) ) ¯ is a minimal prime ideal of ℜ ¯ by Theorem 4.6. Therefore, K ′ ∈ A , as desired.

  2. follows from Theorem 4.5, Theorem 4.6, and the fact that the irreducible components of a topological space are the closed subsets of it.

  3. Since the irreducible components of a topological space cover it, we have Spec L ( ℑ ) = ⋃ K ∈ A ν s ( K ) by (2).

  4. Since Spec L ( ℑ ) = ⋃ K ∈ A ν s ( K ) and φ is surjective, we have Spec ( ℜ ¯ ) = φ ( ⋃ K ∈ A ν s ( K ) ) = ⋃ K ∈ A φ ( ν s ( K ) ) = ⋃ K ∈ A V ℜ ¯ ( Ann ℜ ( K ) ¯ ) using Proposition 2.10.

  5. Since Spec ( ℜ ¯ ) = ⋃ K ∈ A V ℜ ¯ ( Ann ℜ ( K ) ¯ ) , we have

    Spec s ( ℑ ) = ψ − 1 ( Spec ( ℜ ¯ ) ) = ψ − 1 ( ⋃ K ∈ A V ℜ ¯ ( Ann ℜ ( K ) ¯ ) ) = ⋃ K ∈ A ψ − 1 ( V ℜ ¯ ( Ann ℜ ( K ) ¯ ) ) = ⋃ K ∈ A V s ( Ann ℑ ( Ann ℜ ( K ) ) ) = ⋃ K ∈ A V s ( K )

    by Lemma 2.8(1) and [12, Lemma 3.3(b)].

  6. Suppose that ℑ ∈ Spec L ( ℑ ) . Then, ℑ is a secondary submodule of ℑ , and hence, Ann ℜ ( ℑ ) ¯ ∈ Spec ( ℜ ¯ ) . But for any K ∈ A , we have K ⊆ ℑ , and hence, Ann ℜ ( ℑ ) ¯ ⊆ Ann ℜ ( K ) ¯ . Thus, Ann ℜ ( ℑ ) = Ann ℜ ( K ) as Ann ℜ ( K ) ¯ is a minimal prime ideal of ℜ ¯ . This implies that ν s ( K ) = ν s ( ℑ ) = Spec L ( ℑ ) by Lemma 2.4(3). So, we obtain ν s ( K ) = Spec L ( ℑ ) for any K ∈ A . By (2), we have ℬ = { ν s ( K ) ∣ K ∈ A } = { Spec L ( ℑ ) } , which is the set of all irreducible components of Spec L ( ℑ ) , as needed.□

Recall that a topological space Z is said to be a T 0 -space if the closures of distinct points are distinct. Note that Spec L ( ℑ ) for an ℜ -module ℑ is not always a T 0 -space. For example, if ℑ is a vector space over a filed F with dim ( ℑ ) > 1 , then there exist N 1 , N 2 ∈ Spec s ( ℑ ) ⊆ Spec L ( ℑ ) such that N 1 ≠ N 2 . But by Corollary 3.3, the Zariski topology on Spec L ( ℑ ) is the trivial topology. Since the trivial topology on any set X is a T 0 -space if and only if ∣ X ∣ ≤ 1 , the Zariski topology on Spec L ( ℑ ) is not a T 0 -space. In fact, if ℑ is a vector space, then the Zariski topology on Spec L ( ℑ ) is a T 0 -space if and only if dim ( ℑ ) ≤ 1 .

Theorem 4.9

Let ℑ be an ℜ -module. Then, Spec L ( ℑ ) is a T 0 -space if and only if ∣ Spec p L ( ℑ ) ∣ ≤ 1 for any p ∈ Spec ( ℜ ) .

Proof

Suppose that Spec L ( ℑ ) is a T 0 -space, and let K , K ′ ∈ Spec p L ( ℑ ) . Then, Ann ℜ ( K ) = Ann ℜ ( K ′ ) , and thus, Cl ( { K } ) = ν s ( K ) = ν s ( K ′ ) = Cl ( { K ′ } ) . Therefore, K = K ′ . Conversely, let K , K ′ ∈ Spec L ( ℑ ) be such that K ≠ K ′ and assume by way of contradiction that Cl ( { K } ) = Cl ( { K ′ } ) . Then, ν s ( K ) = ν s ( K ′ ) , and hence, Ann ℜ ( sec ( K ) ) = Ann ℜ ( sec ( K ′ ) ) ∈ Spec ( ℜ ) . By hypothesis, K = K ′ , which is a contradiction. This means that Cl ( { K } ) ≠ Cl ( { K ′ } ) . Therefore, Spec L ( ℑ ) is a T 0 -space.□

The following is an easy consequence of Proposition 2.6 and Theorem 4.9.

Corollary 4.10

Let ℑ be an ℜ -module. Then, the following statements are equivalent:

  1. Spec L ( ℑ ) is a T 0 -space.

  2. ∣ Spec p L ( ℑ ) ∣ ≤ 1 for any p ∈ Spec ( ℜ ) .

  3. If ν s ( K 1 ) = ν s ( K 2 ) , then K 1 = K 2 for any K 1 , K 2 ∈ Spec L ( ℑ ) .

  4. The natural map φ is injective.

Let Z be a topological space. Recall that Z is said to be a spectral space if it is homeomorphic to the prime spectrum of a ring equipped with the Zariski topology. In Definition 1.1.5 of [16], a space Z is called a spectral space if it satisfies the following conditions:

  1. Z is a T 0 -space.

  2. Z is quasi-compact.

  3. Any irreducible closed subset of Z has a generic point.

  4. The family of all quasi-compact open subsets of Z is closed under finite intersections and forms an open base.

A lot of facts about spectral spaces are included in [16].

Theorem 4.11

Let ℑ be an ℜ -module. If φ is surjective, then Spec L ( ℑ ) is a spectral space if and only if Spec L ( ℑ ) is a T 0 -space.

Proof

By Theorems 3.4, 3.5, and 4.5.□

The following result is obtained by combining Corollaries 2.11, 4.10, and 4.11.

Corollary 4.12

Let ℑ be an ℜ -module. If φ is surjective, then the following statements are equivalent:

  1. Spec L ( ℑ ) is a spectral space.

  2. Spec L ( ℑ ) is a T 0 -space.

  3. ∣ Spec p L ( ℑ ) ∣ ≤ 1 for every p ∈ Spec ( ℜ ) .

  4. φ is injective.

  5. Spec L ( ℑ ) is homeomorphic to Spec ( ℜ ¯ ) under φ .

Let ℑ be an ℜ -module. By Corollary 4.12, the surjectivity of φ implies that Spec L ( ℑ ) is a spectral space if and only if ∣ Spec p L ( ℑ ) ∣ ≤ 1 for any p ∈ Spec ( ℜ ) . In the following proposition, we replace the surjectivity of φ by the finiteness of Spec L ( ℑ ) .

Proposition 4.13

Let ℑ be an ℜ -module such that Spec L ( ℑ ) is a non-empty finite set. Then, Spec L ( ℑ ) is a spectral space if and only if ∣ Spec p L ( ℑ ) ∣ ≤ 1 for every p ∈ Spec ( ℜ ) .

Proof

Since Spec L ( ℑ ) is finite, we have (b) and (d) in Definition 1.1.5 of [16] are satisfied. Now, let F = { a 1 , a 2 , … , a n } be an irreducible closed subset of Spec L ( ℑ ) . Since F is closed, we have F = Cl ( { a 1 } ∪ { a 2 } ∪ … ∪ { a n } ) = Cl ( { a 1 } ) ∪ Cl ( { a 2 } ) ∪ … ∪ Cl ( { a n } ) . Since F is irreducible, we have F = Cl ( { a i } ) for some i ∈ { 1 , … , k } . So every irreducible closed subset of Spec L ( ℑ ) has a generic point, i.e., condition (c) in Definition 1.1.5 of [16] is also satisfied. Now, Spec L ( ℑ ) is a spectral space if and only if Spec L ( ℑ ) is a T 0 -space if and only if ∣ Spec p L ( ℑ ) ∣ ≤ 1 for any p ∈ Spec ( ℜ ) by Theorem 4.9.□

Recall that a topological space Z is said to be a T 1 -space if every singleton subset of Z is closed. Now, we need the next two lemmas to investigate Spec L ( ℑ ) with the Zariski topology from the viewpoint of being a T 1 -space.

Lemma 4.14

Let ℑ be an ℜ -module such that every secondary submodule of ℑ contains a minimal submodule and let K ∈ Spec L ( ℑ ) . Then, the set { K } is closed in Spec L ( ℑ ) if and only if K is a minimal submodule of ℑ and Spec p L ( ℑ ) = { K } , where p = Ann ℜ ( K ) .

Proof

⇒ : Suppose that { K } is closed in Spec L ( ℑ ) . Then, Cl ( { K } ) = { K } , and thus, ν s ( K ) = { K } . By hypothesis, there exists a minimal submodule T of ℑ such that T ⊆ K . Hence, Ann ℜ ( K ) ⊆ Ann ℜ ( T ) . Since T is a minimal submodule of ℑ , we have T ∈ Spec s ( ℑ ) ⊆ Spec L ( ℑ ) by [6, Proposition 1.6]. Thus, T ∈ ν s ( K ) = { K } . This implies that T = K , and thus, K is a minimal submodule of ℑ . Now, if K ′ ∈ Spec p L ( ℑ ) , then Ann ℜ ( K ′ ) = Ann ℜ ( K ) = p , and hence, K ′ ∈ ν s ( K ) = { K } . This implies that Spec p L ( ℑ ) ⊆ { K } . Consequently, Spec p L ( ℑ ) = { K } .

⇐ : Suppose that B ∈ Cl ( { K } ) . Then, B ∈ ν s ( K ) , and thus, Ann ℜ ( K ) ⊆ Ann ℜ ( B ) . Since K is a minimal submodule of ℑ , we have Ann ℜ ( K ) is a maximal ideal of ℜ , which implies that Ann ℜ ( K ) = Ann ℜ ( B ) . So, p = Ann ℜ ( K ) = Ann ℜ ( B ) . But Spec p L ( ℑ ) = { K } . Thus, B = K , and hence, Cl ( { K } ) ⊆ K . Therefore, { K } is closed in Spec L ( ℑ ) , as desired.□

Lemma 4.15

Let ℑ be an ℜ -module and p ∈ Spec ( ℜ ) . If K 1 , K 2 ∈ Spec p L ( ℑ ) , then K 1 + K 2 ∈ Spec p L ( ℑ ) .

Proof

First, note that Ann ℜ ( K 1 + K 2 ) = Ann ℜ ( K 1 ) ∩ Ann ℜ ( K 2 ) = Ann ℜ ( K 1 ) ∩ Ann ℜ ( K 2 ) = p . Now, we show that K 1 + K 2 is a secondary submodule of ℑ . So let r ∈ ℜ , and suppose that r ∉ Ann ℜ ( K 1 + K 2 ) = Ann ℜ ( K 1 ) = Ann ℜ ( K 2 ) = p . Since K 1 and K 2 are the secondary submodules of ℑ , we have r K 1 = K 1 and r K 2 = K 2 . This implies that r ( K 1 + K 2 ) = K 1 + K 2 . Now, it remains to prove that Ann ℜ ( sec ( K 1 + K 2 ) ) = Ann ℜ ( K 1 + K 2 ) . By [3, Proposition 2.1(h)], sec ( K 1 + K 2 ) ⊆ Ann ℑ ( Ann ℜ ( K 1 + K 2 ) ) , and hence, p = Ann ℜ ( K 1 + K 2 ) ⊆

Ann ℜ ( Ann ℑ ( Ann ℜ ( K 1 + K 2 ) ) ) ⊆ Ann ℜ ( sec ( K 1 + K 2 ) ) .

Since K 1 ⊆ K 1 + K 2 , we have sec ( K 1 ) ⊆ sec ( K 1 + K 2 ) , and hence,

Ann ℜ ( sec ( K 1 + K 2 ) ) ⊆ Ann ℜ ( sec ( K 1 ) ) = p = Ann ℜ ( K 1 + K 2 ) ,

as needed.□

Theorem 4.16

Let ℑ be an ℜ -module such that every secondary submodule of  ℑ contains a minimal submodule. Then, Spec L ( ℑ ) is a T 1 -space if and only if Min ( ℑ ) = Spec L ( ℑ ) , where Min ( ℑ ) is the set of all minimal submodules of ℑ .

Proof

⇒ : Suppose that Spec L ( ℑ ) is a T 1 -space. By [6, Proposition 1.6], Min ( ℑ ) ⊆ Spec s ( ℑ ) ⊆ Spec L ( ℑ ) . Now, let K ∈ Spec L ( ℑ ) . Since Spec L ( ℑ ) is a T 1 -space, we have { K } is closed. By Lemma 4.14, we obtain K ∈ Min ( ℑ ) .

⇐ : Suppose that Min ( ℑ ) = Spec L ( ℑ ) , and let K ∈ Spec L ( ℑ ) . It is enough to show that { K } is closed in Spec L ( ℑ ) . Let p = Ann ℜ ( K ) and H ∈ Spec p L ( ℑ ) . By Lemma 4.15, K + H ∈ Spec p L ( ℑ ) ⊆ Min ( ℑ ) , and thus, K + H ∈ Min ( ℑ ) . But H ⊆ H + K and W ⊆ H + K . So H = H + K = K , and so Spec p L ( ℑ ) = { K } . By Lemma 4.14, { K } is closed in Spec L ( ℑ ) .□

5 Conclusion

In this article, we introduced the notion of the secondary cotop module, which is a generalization of the cotop module.

For this, we define the variety of any submodule N of an ℜ -module ℑ by ν s * ( N ) = { K ∈ Spec L ( ℑ ) ∣ sec ( K ) ⊆ N } and we set Θ s * ( ℑ ) = { ν s * ( N ) ∣ N ≤ ℑ } . Then, we say that ℑ is a secondary cotop module if Θ s * ( ℑ ) is closed under finite unions. It is clear that ν s * ( ℑ ) = Spec L ( ℑ ) , ν s * ( { 0 } ) = ∅ , and ⋂ j ∈ Δ ν s * ( N j ) = ν s * ( ⋂ j ∈ Δ N j ) for any family of submodules { N j } j ∈ Δ and any non-empty index set Δ (Theorem 2.1). So if ℑ is a secondary cotop module, then there exists a topology on Spec L ( ℑ ) having Θ s * ( ℑ ) as a collection of closed sets, and this topology is called the quasi-Zariski topology on Spec L ( ℑ ) . Next, we introduce another variety for any N ≤ ℑ by ν s ( N ) = { K ∈ Spec L ( ℑ ) ∣ Ann ℜ ( N ) ⊆ Ann ℜ ( K ) } and prove that there always exists a topology on Spec L ( ℑ ) having Θ s ( ℑ ) = { ν s ( N ) ∣ N ≤ ℑ } as the collection of closed sets. We call this topology the Zariski topology on Spec L ( ℑ ) or simply Sℒ -topology (Theorem 2.3). We provide some relationships between the varieties ν s * ( N ) , ν s ( N ) , V s * ( N ) , and V s ( N ) for any submodule N of an ℜ -module ℑ and conclude that every secondary cotop module is a cotop module (Lemma 2.4 and Corollary 2.5). In addition, using these relations, we see that the Zariski topology on Spec s ( ℑ ) for an ℜ -module ℑ is a topological subspace of Spec L ( ℑ ) equipped with the Sℒ -topology. Next, for an ℜ -module ℑ , we introduce the map φ : Spec L ( ℑ ) → Spec ( ℜ ∕ Ann ℜ ( ℑ ) ) by φ ( K ) = Ann ℜ ( K ) ∕ Ann ℜ ( ℑ ) to obtain some relations between the properties of Spec L ( ℑ ) and Spec ( ℜ ∕ Ann ℜ ( ℑ ) ) . For example, we relate the connectedness of Spec L ( ℑ ) to the connectedness of both Spec ( ℜ ∕ Ann ℜ ( ℑ ) ) and Spec s ( ℑ ) using φ and ψ (Theorem 2.12). We show that the secondary-like spectrums of two modules are homeomorphic if there exists a module isomorphism between these modules (Corollary 2.15). We introduce a base for the Zariski topology on Spec L ( ℑ ) (Theorem 3.1). Also, we prove that the surjectivity of φ implies that the basic open sets are quasi-compact and conclude that Spec L ( ℑ ) is quasi-compact (Theorem 3.4). We investigate the irreducibility of Spec L ( ℑ ) with respect to the Sℒ -topology. In particular, it is shown that if φ is surjective, then every irreducible closed subset of Spec L ( ℑ ) has a generic point and we determine exactly the set of all irreducible components of Spec L ( ℑ ) (Theorem 4.5 and Corollary 4.8). Furthermore, we study Spec L ( ℑ ) from the viewpoint of being a T 0 -space, a T 1 -space, and a spectral space.

Acknowledgement

The authors wish to thank sincerely the referees for their valuable comments and suggestions.

  1. Funding information: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

  2. Conflict of interest: The authors state no conflicts of interest.

  3. Data availability statement: No data were used for the research described in the article.

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Received: 2023-06-21
Revised: 2023-12-20
Accepted: 2024-03-07
Published Online: 2024-04-05

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
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  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
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  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
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  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
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  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0005
Keywords for this article
secondary-like spectrum; Zariski topology; second spectrum
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