@ Appl. Gen. Topol. 23, no. 2 (2022), 333-343 doi:10.4995/agt.2022.17427 © AGT, UPV, 2022 Zariski topology on the spectrum of fuzzy classical primary submodules Phakakorn Panpho a and Pairote Yiarayong b a Major of Physics, Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok 65000, Thailand (phakakorn.p@psru.ac.th) b Department of Mathematics, Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok 65000, Thailand (pairote0027@hotmail.com) Communicated by S. Romaguera Abstract Let R be a commutative ring with identity and M a unitary R-module. The fuzzy classical primary spectrum Fcp.spec(M) is the collection of all fuzzy classical primary submodules A of M, the recent generalization of fuzzy primary ideals and fuzzy classical prime submodules. In this paper, we topologize FM(M) with a topology having the fuzzy primary Zariski topology on the fuzzy classical primary spectrum Fcp.spec(M) as a subspace topology, and investigate the properties of this topological space. 2020 MSC: 08A72; 54B35; 13C05; 13C13; 13C99; 16D10; 03E72; 16N80; 16W50. Keywords: Zariski topology; classical primary submodule; fuzzy classical primary submodule; fuzzy classical primary spectrum; fuzzy pri- mary ideal. 1. Introduction Throughout this paper all rings are commutative with identity and all mod- ules are unitary. The fuzzy classical primary submodule in module theory plays crucial role in algebra. The concept of fuzzy classical primary submodule which is a generalization of fuzzy primary ideals and fuzzy classical prime sub- modules. The Zariski topology on the prime spectrum of an R-module have Received 01 April 2022 – Accepted 06 July 2022 http://dx.doi.org/10.4995/agt.2022.17427 https://orcid.org/0000-0003-1525-9536 P. Panpho and P. Yiarayong been introduced by Lu [13] and these have been studied by several authors [1, 2, 6, 7, 8, 9, 15, 16]. In 2008, Ameri and Mahjoob [3] investigated some properties of the Zarisky topology of prime L-submodules. As it is well known that Ameri and Mahjoob in 2009 [4] introduced the notion of the Zarisky topol- ogy of prime fuzzy hyperideals. In 2013, Darani and Motmaen [10] introduced and studied the concept of the Zariski topology on the spectrum of graded clas- sical prime submodules and these have been studied by several authors [18]. The concept of Zariski topology on prime fuzzy submodules was introduced Ameri and Mahjoob [5] in 2017. In 2021, Goswami and Saikia [11] gave the concept of the spectrum of weakly prime submodules and investigated related properties. In this paper, we rely on the fuzzy classical primary submodules, and then introduce and study a new topology on the fuzzy classical primary spectrum Fcp.spec(M) is the collection of all fuzzy classical primary submodules A of M, which generalizes the Zariski topology of fuzzy prime submodule, called fuzzy primary Zariski topology and investigate several properties of the topology. 2. Basic definitions and preliminary results In this section, a brief overview of the concepts of fuzzy sets and fuzzy modules are required in this study. A function µ : M → [0, 1] is called a fuzzy set [17] of a non empty set M. The concept of fuzzy ideals of a ring was introduced in [12] as a generalization of the notion of fuzzy subrings. Definition 2.1 ([12]). A fuzzy set µ of a ring R is called a fuzzy ideal of R if (1) µ(ab) ≥ µ(a) ∨µ(b) for all a,b ∈ R; (2) µ(a− b) ≥ µ(a) ∧µ(b) for all a,b ∈ R. The set of all fuzzy sets ( fuzzy ideals) of R is denoted by FS(R) (FI(R)). Let µ be a fuzzy set of a ring R. The radical of µ is denoted by <(µ) and is defined by (<(µ)) (r) = ∨ n∈N µ(rn) for every element r ∈ R. Definition 2.2. Let µ be a fuzzy ideal of a ring R. A fuzzy set µ is called a fuzzy primary ideal of R if for every fuzzy ideals ν and η of R with ν�η ≤ µ, then either ν ≤ µ or η ≤<(µ) The concept of fuzzy modules of an R-module M was introduced in [14] as a generalization of the notion of fuzzy ideals. Definition 2.3 ([14]). A fuzzy set A of an R-module M is called a fuzzy module of M if (1) A(0) = 1; (2) A(am) ≥A(m) for all a ∈ R and m ∈ M; (3) A(m−n) ≥A(m) ∧A(n) for all m,n ∈ M. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 334 Zariski topology on the spectrum of fuzzy classical primary submodules Condition (3) of the above definition is equivalent to A(m + n) ≥ A(m) ∧ A(n), and A(m) = A(−m) for all m,n ∈ M. The set of all fuzzy sets (fuzzy modules) of an R-module M is denoted by FS(M) (FM(M)). Theorem 2.4. Let A be a fuzzy set over an R-module R such that A(0) = 1. Then the following conditions are equivalent. (1) A is a fuzzy module of R. (2) A is a fuzzy ideal of R. Proof. By Definition 2.3 it suffices to prove that (2) implies (1). Assume that (2) holds. Let a and b be any elements of R. Since A is a fuzzy ideal of R with A(0) = 1, we have µ(ab) ≥ µ(a)∨µ(b) ≥ µ(b) and µ(a−b) ≥ µ(a)∧µ(b). Therefore, we obtain that A is a fuzzy module of R. � Let N be a non empty subset of an R-module M. For each α ∈ [0, 1), a characteristic function of N is denoted by αCN and is defined as (αCN ) (m) = { 1 ; m ∈ N α ; otherwise. We note that the R-module M can be considered a bipolar fuzzy set of itself and we write M = αCM (R = αCR), i.e., M(m) = 1 for all m ∈ M. In the following theorem, we establish a relationship between bipolar fuzzy modules and submodules of an R-module. Theorem 2.5. Let α be any element of [0, 1) and let M be an R-module. Then the following conditions are equivalent. (1) N is a submodule of M. (2) The characteristic function αCN of N is a fuzzy module over M. Proof. First assume that N is a submodule of M. Since 0 is an element of N, we have (αCN ) (0) = 1. Let m and n be any elements of M and r ∈ R. If m,n ∈ N, then (αCN ) (m) = 1 = (αCN ) (n) and since m−n,rm ∈ N, we have (αCN ) (rm) = 1 = (αCN ) (m) and (αCN ) (m−n) = 1 = (αCN ) (m)∧(αCN ) (n). Otherwise, if m 6∈ N or n 6∈ N, then (αCN ) (m) = α or (αCN ) (m) = α and so we have (αCN ) (m−n) ≥ α = (αCN ) (m) ∧ (αCN ) (n). It is obvious that (αCN ) (rm) ≥ α = (αCN ) (m). Therefore αCN is a fuzzy module over M and hence (1) implies (2). Conversely, assume that (2) holds. Let m and n be any elements of M and r ∈ R such that m,n ∈ N. Set x = rm and y = m − n. Then (αCN ) (x) = (αCN ) (rm) ≥ (αCN ) (m) = 1 and (αCN ) (y) = (αCN ) (m−n) ≥ (αCN ) (m) ∧ (αCN ) (n) = 1 ∧ 1 = 1. Hence we have (αCN ) (x) = 1 and (αCN ) (y) = 1, and so m − n,rm ∈ N. Therefore N is a submodule of M and hence (1) implies (2). � From the above result, we have the following corollary: Corollary 2.6. Let M be an R-module. Then the following conditions are equivalent. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 335 P. Panpho and P. Yiarayong (1) N is a submodule of M. (2) The characteristic function 0CN of N is a fuzzy module over M. Let µ and A be a fuzzy set over a ring R and fuzzy set over an R-module M, respectively. Define the composition µ�A, and product µA respectively as follows: (µ�A) (x) =   ∨ x=rm (µ(r) ∧A(m)) ; if x = rm for some r ∈ R,m ∈ M 0 ; otherwise, and (µA) (x) =   ∨ x= n∑ i rimi n∧ i=1 µ(ri) ∧ n∧ i=1 A(mi) ;x = n∑ i rimi∃ri ∈ R,mi ∈ M 0 ; otherwise. Next, let x be an element of an R-module M and α ∈ (0, 1]. Define the fuzzy set xα over M as follows: xα(a) = { α ; x = a 0 ; otherwise. Then xα is called a fuzzy point or fuzzy singleton. Let A be a fuzzy set over an R-module M. Next, let 〈A〉 denote the intersection of all fuzzy modules over M which contain A. Then 〈A〉 is a fuzzy module over M, called the fuzzy module generated by A. Definition 2.7. Let A and B be any fuzzy sets of an R-module M. For every fuzzy set µ of R define (A : B) and (A : µ), as follows: (A : B) = ∨ {µ ∈FS(R) : µ�B ≤A} and (A : µ) = ∨ {B ∈FS(M) : µ�B ≤A}. 3. Topologies on fuzzy classical primary submodules The given definition of fuzzy classical primary submodule is a generalization of the notion of classical prime and classical primary submodules in module theory. Definition 3.1. Let A be a fuzzy submodule of an R-module M. A fuzzy set A is called a fuzzy classical primary submodule of M if for every elements a and b of R and every element x of M with aζbξxα ∈A, then either aζxα ∈A or bnζ xα ∈A for some positive integer n. We now present the following example satisfying above definition. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 336 Zariski topology on the spectrum of fuzzy classical primary submodules Example 3.2. Let Z be the set of all integers. Suppose M = R = Z is a commutative ring. Define the fuzzy set A of Z as follows: A(x) = { 1 ; if x ∈ 4Z 0 ; if x 6∈ 4Z. Then it is easily seen that A is a fuzzy classical primary submodule of an an R-module M. Let M be an R-module. In the sequel Fcp.spec(M) denotes the set of all fuzzy classical primary submodules of an R-module M. We call Fcp.spec(M), the fuzzy classical primary spectrum of M. For every fuzzy submodule A of M, the fuzzy classical variety of A is denoted by V(A), and is defined as the set of all fuzzy classical primary submodule containing A, i.e., V(A) = {B ∈ Fcp.spec(M) : A≤B}. Theorem 3.3. For any family of fuzzy submodules {Ai}i∈I of an R-module M. Then the following properties hold. (1) V(01) = Fcp.spec(M) and V(M) = ∅. (2) ⋂ i∈I V (Ai) = V (∑ i∈I Ai ) . (3) V (A1) ∪V (A2) = V (A1 ∧A2). Proof. (1). Obvious. (2). Let B be a fuzzy submodule of M such that B ∈ ⋂ i∈I V (Ai). Then we have B ∈V (Ai) for all i ∈ I, i.e., Ai ≤B. Next let x be an element of M. We also consider (∑ i∈I Ai ) (x) = ∨ x= ∑ i∈I xi   ∧ x= ∑ i∈I xi Ai(xi)   ≤ ∨ x= ∑ i∈I xi   ∧ x= ∑ i∈I xi B(xi)   = ∨ x= ∑ i∈I xi B(xi) = B(x) © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 337 P. Panpho and P. Yiarayong ans so ∑ i∈I Ai ≤ B. Thus we have B ∈ V (∑ i∈I Ai ) , which implies that ⋂ i∈I V (Ai) ⊆ V (∑ i∈I Ai ) . On the other hand, let B be a fuzzy submodule of M such that B ∈ V (∑ i∈I Ai ) . It is easy to see that Ai ≤ ∑ i∈I Ai ≤ B, i.e., B ∈ V (Ai) for all i ∈ I. Therefore V (∑ i∈I Ai ) ⊆ ⋂ i∈I V (Ai) and hence ⋂ i∈I V (Ai) = V (∑ i∈I Ai ) . (3). Let B be a fuzzy submodule of M such that B ∈V (A1)∪V (A2). Then we have A1 ≤B or A2 ≤B, it follows that A1∧A2 ≤B. Thus B ∈V (A1 ∧A2) and so V (A1) ∪V (A2) ⊆ V (A1 ∧A2). On the other hand, let B be a fuzzy submodule of M such that B ∈ V (A1 ∧A2). This implies that A1 ∧A2 ≤ B, i.e., A1 ≤B or A2 ≤B. Also, B ∈V (A1) ∪V (A2). Therefore, we obtain that V (A1 ∧A2) ⊆V (A1) ∪V (A2) and hence V (A1) ∪V (A2) = V (A1 ∧A2). � Corollary 3.4. Let µ and ν be any fuzzy ideal of a ring R. Then V (µ�M)∪ V (ν �M) = V (µ�ν �M). Set X = Fcp.spec(M). For every fuzzy submodule A of an R-module M we define E(A) and τ as follows: E(A) = X −V (A) and τ = {E(A) : A∈FM(M)}. In the next theorem we will show that the pair (X,τ) is a topological space. Theorem 3.5. Let M be an R-module. Then the following statements hold: (1) The pair (X,τ) is a topological space. (2) X is a T0 topological space. Proof. 1. Since V(01) = X and V(M) = ∅, we have E(01) = X −X = ∅ and E(M) = X −∅ = X, .i.e., ∅,X ∈ τ. 2. Let A and B be any fuzzy submodules of M. Thus by Theorem 3.3(3), we have E(A) ∩E(B) = (X −V (A)) ∩ (X −V (B)) = X ∩ ( V (A)′ ∩V (B)′ ) = X ∩ (V (A) ∪V (B))′ = X −V (A∧B) = E(A∧B). © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 338 Zariski topology on the spectrum of fuzzy classical primary submodules 3. For any family of fuzzy submodules {Ai}i∈I of M. Then by Theorem 3.7(2), we have ⋃ i∈I E(Ai) = ⋃ i∈I (X −V (Ai)) = X − ⋂ i∈I V (Ai) = X −V (∑ i∈I Ai ) = E (∑ i∈I Ai ) . 2 and 3 show that τ is closed under arbitrary union and finite intersection. Thus the pair (X,τ) satisfies in axioms of a topological space. Therefore we have (X,τ) is a topological space. (2) Let A and B be two distinct points of X. If A 6≤ B, then obviously B ∈E(A) and A 6∈ E(A) showing that X is a T0 topological space. � In this case, the topology τ on X is called the fuzzy primary Zariski topology. For every fuzzy submodule A of M, the set V∗(A) = { B ∈ Fcp.spec(M) : √ (A : M) ≤ √ (B : M) } . Then we have the following lemma. Lemma 3.6. Let A and B be any fuzzy submodules of an R-module M. If A≤B, then V∗ (B) ≤V∗ (A). Proof. Let C be a fuzzy submodule of M such that C ∈V∗ (B). Then we have√ (B : M) ≤ √ (C : M). Since A ≤ B, we have √ (A : M) ≤ √ (B : M), i.e.,√ (A : M) ≤ √ (C : M). Therefore C ∈V∗ (A) and hence V∗ (B) ≤V∗ (A). � Then we have the next results. Theorem 3.7. For any family of fuzzy submodules {Ai}i∈I of an R-module M. Then the following properties hold. (1) V∗(01) = Fcp.spec(M) and V∗(M) = ∅. (2) ⋂ i∈I V∗ (Ai) = V∗ (∑ i∈I (Ai : M) �M ) . (3) V∗ (A1) ∪V∗ (A2) = V∗ (A1 ∧A2). Proof. (1). Obvious. (2). Let B be a fuzzy submodule of M such that B ∈ ⋂ i∈I V∗ (Ai). Then we have B ∈V∗ (Ai) for all i ∈ I, i.e., √ (Ai : M) ≤ √ (B : M). Since (Ai : M)� M ≤ √ (Ai : M) � M ≤ √ (B : M) � M, we have ∑ i∈I (Ai : M) � M ≤ © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 339 P. Panpho and P. Yiarayong √ (B : M) �M, it follows that,√√√√((∑ i∈I (Ai : M) �M ) : M ) ≤ √(√ (B : M) �M : M ) ≤ √√ (B : M) = √ (B : M). It is easy to see that B ∈ V∗ (∑ i∈I (Ai : M) �M ) and so ⋂ i∈I V∗ (Ai) ⊆ V∗ (∑ i∈I (Ai : M) �M ) . On the other hand, let B be a fuzzy submodule of M such that B ∈V∗ (∑ i∈I (Ai : M) �M ) . Thus we have √√√√((∑ i∈I (Ai : M) �M ) : M ) ≤ √ (B : M). Clearly, we have (((Ai : M) �M) : M) = (Ai : M) for all i ∈ I. Also for each i ∈ I, we obtain that√ (Ai : M) = √ (((Ai : M) �M) : M) ≤ √√√√((∑ i∈I (Ai : M) �M ) : M ) ≤ √ (B : M) = √ (B : M). Therefore we obtain that B ∈ ⋂ i∈I V∗ (Ai) and hence V∗ (∑ i∈I (Ai : M) �M ) ⊆⋂ i∈I V∗ (Ai). (3). Let B be a fuzzy submodule of M such that B ∈ V∗ (A1) ∪V∗ (A2). Then we have √ (A1 : M) ≤ √ (B : M) or √ (A2 : M) ≤ √ (B : M). If√ (A1 : M) ≤ √ (B : M), then √ (A1 ∧A2 : M) ≤ √ (A1 : M) ≤ √ (B : M), it follows that, B ∈ V∗ (A1 ∧A2). Similarly, if √ (A2 : M) ≤ √ (B : M), then B ∈ V∗ (A1 ∧A2). On the other hand, let B be a fuzzy submodule of M such that B ∈ V∗ (A1 ∧A2). Then √ (A1 ∧A2 : M) ≤ √ (B : M). Since A1∧A2 ≤A1 and A1∧A2 ≤A2, we have √ (A1 : M) ≤ √ (A1 ∧A2 : M) and√ (A2 : M) ≤ √ (A1 ∧A2 : M), which implies that,√ (A1 : M) � √ (A2 : M) ≤ √ (A1 ∧A2 : M). © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 340 Zariski topology on the spectrum of fuzzy classical primary submodules Now since √ (B : M) is prime and √ (A1 : M) � √ (A2 : M) ≤ √ (B : M), it follows that √ (A1 : M) ≤ √ (B : M) or √ (A2 : M) ≤ √ (B : M). Clearly, we have B ∈ V∗ (A1) or B ∈ V∗ (A2), i.e., B ∈ V∗ (A1) ∪V∗ (A2). Therefore V∗ (A1 ∧A2) ≤V∗ (A1)∪V∗ (A2) and hence V∗ (A1)∪V∗ (A2) = V∗ (A1 ∧A2). � For every fuzzy submodule A of an R-module M we define E∗(A) and τ∗ as follows: E∗(A) = X −V∗ (A) and τ∗ = {E∗(A) : A∈FM(M)}. In the next theorem we will show that the pair (X,τ∗) is a topological space. Theorem 3.8. Let M be an R-module. Then the following statements hold: (1) The pair (X,τ∗) is a topological space. (2) X is a T0 topological space. Proof. The proof follows from Theorem 3.5. � For any R-module M and A,B ∈FM(M) we have the next result. Proposition 3.9. Let A and B be any fuzzy submodules of an R-module M. If √ (A : M) = √ (B : M), then V∗ (A) = V∗ (B). Moreover, the converse is true if both A and B are classical primary. Proof. Let A and B be any fuzzy submodules of M such that √ (A : M) =√ (B : M). Next let C be a fuzzy submodule of M such that C ∈ V∗ (A). Then we have √ (A : M) ≤ √ (C : M), i.e., √ (B : M) ≤ √ (C : M). Thus C ∈ V∗ (B) and so V∗ (A) ⊆ V∗ (B). Similarly, we obtain that V∗ (B) ⊆ V∗ (A). For the converse, suppose that A,B ∈ FM(M) is classical primary and V∗ (A) = V∗ (B). Since A ∈ V∗ (A) ,B ∈ V∗ (B) and V∗ (A) = V∗ (B), we have √ (B : M) ≤ √ (A : M) and √ (A : M) ≤ √ (B : M). Therefore, we obtain that √ (A : M) = √ (B : M). � For a fuzzy prime ideal p of R, by Fcp.specp(M) we mean the set of all A∈FM(M) such that √ (A : M) = p. In other words Fcp.specp(M) = { A∈ Fcp.spec(M) : √ (A : M) = p } . Theorem 3.10. Let µ and A be any fuzzy ideal and any fuzzy submodule of R and M, respectively. Then the following properties hold. (1) V∗(A) = ⋃ √ (A:M)≤p Fcp.specp(M). (2) V∗ (µm �M) = V (µn �M) for some positive integers m,n. (3) V (√ (A : M) �M ) ⊆V∗ (A) ⊆V∗ ((A : M) �M). Proof. (1). Let B be a fuzzy submodule of M such that B ∈V∗ (A). Then we have √ (A : M) ≤ √ (B : M) = p and so © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 341 P. Panpho and P. Yiarayong B ∈ Fcp.specp(M) ⊆ ⋃ √ (A:M)≤p Fcp.specp(M). It is easy to see that V∗(A) ⊆ ⋃ √ (A:M)≤p Fcp.specp(M). On the other hand, let B be a fuzzy submodule of M such that B ∈ ⋃ √ (A:M)≤p Fcp.specp(M). Thus there exists a fuzzy prime ideal p of R such that √ (A : M) ≤ p and B ∈ Fcp.specp(M). Clearly, we have √ (B : M) = p, i.e., √ (A : M) ≤ √ (B : M), it follows that, B ∈V∗ (A). Therefore we obtain that ⋃ √ (A:M)≤p Fcp.specp(M) ⊆ V∗(A) and hence V∗(A) = ⋃ √ (A:M)≤p Fcp.specp(M). (2). Let B be a fuzzy submodule of M such that B ∈ V (µn �M). Then we have µn � M ≤ B, i.e., √ (µm �M : M) ≤ √ (B : M). This implies that B ∈ V∗ (µm �M) and so V (µn �M) ⊆ V∗ (µm �M). On the other hand, let B be a fuzzy submodule of M such that B ∈ V∗ (µm �M). Thus√ (µm �M : M) ≤ √ (B : M). Obviously, µm ≤ (µm �M : M). Since√ (µm �M : M) ≤ √ (B : M) and µm ≤ (µn �M : M), we have µm ≤√ (B : M), which implies that, µmt � M ≤ B. It is easy to see that B ∈ V (µn �M). Therefore V∗ (µm �M) ⊆V (µn �M) and hence V∗ (µm �M) = V (µn �M). (3). Let B be a fuzzy submodule of M such that B ∈V∗ (A). Then we have√ (A : M) ≤ √ (B : M). Since (A : M) �M≤A, we have√ ((A : M) �M : M) ≤ √ (A : M) ≤ √ (B : M). This implies that B ∈V∗ ((A : M) �M) and so V∗ (A) ⊆V∗ ((A : M) �M). Next, let B be a fuzzy submodule of M such that B ∈ V (√ (A : M) �M ) . Thus √ (A : M)�M≤B. Obviously, √ (A : M) ≤ (B : M). Since (B : M) ≤√ (B : M), we have √ (A : M) ≤ √ (B : M), which implies that, B ∈ V∗ (A). 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