Int. J. Anal. Appl. (2023), 21:84 On M∗-Irresolute Topological Rings Shallu Sharma1,∗, Naresh Digra1, Pooja Saproo1, Tsering Landol2 1Department of Mathematics, University of Jammu, Jammu, India 2Department of Mathematics, Cluster University of Jammu, Jammu, India ∗Corresponding author: shallujamwal09@gmail.com Abstract. The main aim of this paper is to introduce and study the new notions namely M∗-irresolute topological rings and M∗-irresolute topological R-modules by virtue of M∗-open sets. Examples of an M∗-irresolute topological ring and module have been put forth. Further, we provide several fundamental properties and characterizations of M∗-irresolute topological rings and M∗-irresolute topological R- modules. In addition, we shall define boundedness in these two structures and present several results on them. 1. Introduction Although topological rings are useful in many branches of mathematics, they are also fascinating on their own. Since the 1940s, the theory of topological rings has been extensively developed, but primarily within the broader concept of a topological module. L.S. Pontryagin obtained one of the first fundamental results in the theory of topological rings in the classification of locally compact skew fields, which was included in his famous book [14] on topological groups. Some topological ring and module properties have also been noted in books [2, 11]. In-depth study has also been done in the last 50 years in the area of normed and Banach algebras, which constitute one of the most significant classes of topological rings (see, for example [5–8,13]). Besides, the theory of topological rings have been thoroughly investigated in a number of review papers and monographs (see, for example [1,9,10,15–18]). In 2016, A. Devika and A. Thilagavathi [4] introduced a new class of sets in topological spaces called M∗-open sets and studied some of its properties. By continuing the study of M∗-open sets and topological rings, in this paper we introduce Received: Jun. 12, 2023. 2020 Mathematics Subject Classification. 54H13, 16W80, 16W99. Key words and phrases. M∗-open sets; M∗-irresolute topological rings; M∗-irresolute topological R-modules. https://doi.org/10.28924/2291-8639-21-2023-84 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-84 2 Int. J. Anal. Appl. (2023), 21:84 a new class of topological rings and modules called M∗-irresolute topological rings and M∗-irresolute topological R-modules. 2. Preliminaries This section covers some fundamental definitions that will be utilized in the next sections. Through- out this section, A and A′ will represent two topological spaces with topologies τ and τ′ respectively, on which no separation axioms are imposed. The notations Int(U) and Cl(U) stand for the interior and closure of a subset U of topological space A, respectively. Definition 2.1. [3] Let (A,τ) be a topological space and U ⊆ A. A point x ∈ A is said to be a θ-interior point of U if there exists an open set O containing x such that O ⊆ Cl(O) ⊆ U. The set of all θ-interior points of U is said to be the θ-interior of U and is denoted by Intθ(U). Lemma 2.1. [3] For a subset U of a topological space (A,τ), the following statements are true: (1) Intθ(U) is the union of all open sets of A whose closures are contained in U. (2) U is θ-open if and only if U = Intθ(U). Definition 2.2. [4] Let (A,τ) be a topological space. Then a subset U of a space (A,τ) is said to be, (1) an M∗-open set if U ⊆ Int(Cl(Intθ(U))). (2) an M∗-closed set if U ⊇ Cl(Int(Clθ(U))). The complement of an M∗-open set is called an M∗-closed. We denote the family of all M∗-open subsets (closed subsets) of A is denoted by τM∗ = M∗-O(A) (M∗-C(A)). Lemma 2.2. [4] Let (A,τ) be a topological space. Then the following assertions are true: (1) The arbitrary union of M∗-open sets is M∗-open. (2) The arbitrary intersection of M∗-closed sets is M∗-closed. Lemma 2.3. [4] For a topological space (A,τ) the family of all M∗-open sets of A forms a topology for A. Definition 2.3. [4] Let U be a subset of a space (A,τ). Then, (1) the intersection of all M∗-closed sets containing U is called the M∗-closure of U and is denoted by ClM∗(U) (sometimes by 〈U〉A). (2) the union of all M∗-open sets contained in U is called the M∗-interior of U and is denoted by IntM∗(U). Theorem 2.1. [4] Let U be a subset of a topological space (A,τ). Then the following statements hold: Int. J. Anal. Appl. (2023), 21:84 3 (1) U is an M∗-open set if and only if U = IntM∗(U). (2) U is an M∗-closed set if and only if U = ClM∗(U). Theorem 2.2. [4] Let U and V be subsets of a topological space (A,τ). Then the following statements are true: (1) ClM∗(A−U) = A− IntM∗(U). (2) IntM∗(A−U) = A−ClM∗(U). (3) If U ⊆ V then ClM∗(U) ⊆ ClM∗(V ) and IntM∗(U) ⊆ IntM∗(V ). (4) ClM∗(ClM∗(U)) = ClM∗(U) and IntM∗(IntM∗(U)) = IntM∗(U). Definition 2.4. [4] A subset U of a topological space (A,τ) is said to be a M∗-neighborhood (Briefly M∗-nbd) of a point x ∈ A if there exists an M∗-open set V such that x ∈ V ⊆ U. The family of M∗-neighborhoods of x ∈A is called the M∗-neighborhood system of x and denoted by M∗-Nx. Theorem 2.3. [4] A subset U of a topological space (A,τ) is said to be M∗-open if and only if it is M∗-neighborhood for every point x ∈ U. Definition 2.5. [12] A mapping g : (A,τ) → (A′,τ′) is called M∗-irresolute at element x ∈A if for any M∗-open set U in A′ containing g(x) there exists an M∗-open set V in A containing x such that g(V ) ⊆ U. Theorem 2.4. [12] Let g : A→A′ be a function. Then the following statements are equivalent. (1) g is M∗-irresolute. (2) For each x ∈A and each M∗-neighborhood U of g(x) in A′, there is an M∗-neighborhood V of x in A such that g(V ) ⊆ U. (3) The inverse image of every M∗-closed subset of A′ is an M∗-closed subset of A. (4) The inverse image of every M∗-open subset of A′ is an M∗-open subset of A. Definition 2.6. [12] Consider two topological spaces (A,τ) and (A′,τ′). A function g : A → A′ is called: (1) M∗-continuous if g−1(U) is M∗-open set in A for every open set U in A′. (2) pre-M∗-open if g(U) is M∗-open set in A′ for every M∗-open set U in A. (3) M∗-homeomorphism if g is bijective, M∗-irresolute and pre-M∗-open. Definition 2.7. [12] Let (A,τ) be a topological space. Then A is said to be M∗-compact if each of its cover by M∗-open sets has finite subcover. A subset S ⊆ A is said to be M∗-compact if each cover of S by M∗-open sets of A has a finite subcover. Definition 2.8. [1] Let R be a ring and E be an R-module, S ⊆ E and Q ⊆ R. If Q.S ⊆ S, then the subset S is called Q-stable. 4 Int. J. Anal. Appl. (2023), 21:84 3. M∗-Irresolute topological rings Definition 3.1. An M∗-irresolute topological ring (R, +, ·,τ) is a ring (R, +, ·) endowed with some topology τ such that the following conditions are satisfied: (C1) for each r1, r2 ∈ R and M∗-open set U in R containing r1 + r2, there exist M∗-open sets U1 and U2 in R containing r1 and r2 respectively, such that U1 + U2 ⊆ U; (C2) for each r ∈ R and M∗-open set U in R containing −r, there exists M∗-open set U1 in R containing r such that −U1 ⊆ U; (C3) for each r1, r2 ∈R and M∗-open set U in R containing r1.r2, there exist M∗-open sets U1 and U2 in R containing r1 and r2 respectively, such that U1.U2 ⊆ U. Remark 3.1. It is easy to verify that conditions (C1) and (C2) are equivalent to the following condition: For each r1, r2 ∈R and each M∗-open set U in R containing r1 −r2, there exist M∗-open sets U1 and U2 in R containing r1 and r2, respectively, such that U1 −U2 ⊆ U. Example 3.1. Consider Z4, the ring of integers modulo 4. Define topology τ on Z4 as τ = {∅,{0},{1},{0, 1},{1, 3},{0, 1, 3},{0, 2},{0, 1, 2},Z4}, τM∗ = {∅,{0, 2},{1, 3},Z4}. Then, (Z4,⊕4,�4,τ) is an M∗-irresolute topological ring. Example 3.2. Consider the ring R = (Z3,⊕3,�3). Let τ = {∅,{0, 1},Z3}, then τM∗ = {∅,Z3}. Thus, (Z3,⊕3,�3,τ) is an M∗-irresolute topological ring. Example 3.3. Let (R, +, ·) be any ring and suppose τ be a discrete or indiscrete topology on R . Then (R, +, ·,τ) is always an M∗-irresolute topological ring. Definition 3.2. Let (R, +, ·,τ) be an M∗-irresolute topological ring. A left R-module E is called an M∗-irresolute topological left R-module if on E is specified a topology such that the following conditions are satified: (M1) for every m,n ∈ E and M∗-open set U in E containing m + n, there exist M∗-open sets U1 and U2 in E containing m and n respectively, such that U1 + U2 ⊆ U; (M2) for every m ∈ E and M∗-open set U in E containing −m, there exists an M∗-open set U1 in E containing m such that −U1 ⊆ U; (M3) for every r ∈R and m ∈E and M∗-open set U in E containing r.m, there exist M∗-open set U1 in R containing r and M∗-open set U2 in E containing m such that U1.U2 ⊆ U. Remark 3.2. In a similar manner, M∗-irresolute topological right R-modules over an M∗-irresolute topological ring can be investigated. Any M∗-irresolute topological ring R is both an M∗-irresolute topological left R-module and an M∗-irresolute topological right R-Module. Remark 3.3. Hereafter, by an M∗-irresolute topological R-module (unless otherwise asserted), we mean an M∗-irresolute topological left R-module. Int. J. Anal. Appl. (2023), 21:84 5 Example 3.4. Consider the ring R = (Z4,⊕4,�4) with topology τ = {∅,{0},{1},{0, 1},{1, 3},{0, 1, 3},{0, 2},{0, 1, 2},Z4}, then τM∗ = {∅,{0, 2},{1, 3},Z4}. Therefore, R = (Z4,⊕4,�4,τ) is an M∗-irresolute topological ring (see Example 3.1). Let τ′ be the indiscrete topology on the ring ({0, 2},⊕4,�4), τ′M∗ = {∅,{0, 2}}. Then, left R-module {0, 2} with τ′ is an M∗-irresolute topological left R-module. Note 3.1. By I(R), we denote the set of all invertible elements in an M∗-irresolute topological ring R. Theorem 3.1. Consider an M∗-irresolute topological ring R, an M∗-irresolute topological R-module E, a ∈R, m ∈E, and S be a subset in R, T a subset in E. Then the following statements are true: (1) the mapping Ωa : E → E, where Ωa(x) = a.x, x ∈ E, is an M∗-irresolute mapping of the topological space E into itself. (2) the mapping Ωm : R → E, where Ωm(x) = x.m, x ∈ R is an M∗-irresolute mapping of the topological space R to the topological space E. (3) 〈S.T〉E ⊇〈S〉R.〈T〉E. Proof. (1) Let U be any M∗-open set in E containing Ωa(x) = a.x, where x ∈ E is arbitrary. Then, by Definition 3.2, there exist M∗-open set U1 in R containing a and M∗-open set U2 in E containing x, such that U1.U2 ⊆ U. Thus, Ωa(U2) = a.U2 ⊆ U1.U2 ⊆ U. This proves that Ωa is an M∗-irresolute mapping. (2) Let x ∈ R and let U be any M∗-open set in E containing Ωm(x) = x.m. Then, by Definition 3.2, there exist M∗-open set U1 in R containing x and M∗-open set U2 in E containing m, such that U1.U2 ⊆ U. Thus, Ωm(U1) = U1.m ⊆ U1.U2 ⊆ U. This proves that Ωm is an M∗-irresolute mapping. (3) Let x ∈ 〈S〉R.〈T〉E and let U be an M∗-open set in E containing x. Then x = s.t, where s ∈ 〈S〉R and t ∈ 〈T〉E, and hence, there exist M∗-open set U1 in R containing s and M∗-open set U2 in E containing t, such that U1.U2 ⊆ U. By virtue of the fact that U1 ∩S 6= ∅ and U2 ∩T 6= ∅, elements s1 ∈ U1 ∩S and t1 ∈ U2 ∩T can be found. Thus, s1.t1 ∈ S.T and s1.t1 ∈ U1.U2 ⊆ U, that is (S.T ) ∩U 6= ∅. Consequently, 〈S.T〉E ⊇〈S〉R.〈T〉E. � Theorem 3.2. Let R be an M∗-irresolute topological ring, a ∈R, and let S and T be subsets in R. Then the following statements are true: (1) the mappings Σa : R→R and Σ′a : R→R, where Σa(x) = x.a and Σ′a(x) = a.x for x ∈R, are M∗-irresolute mappings of the topological space R into itself; (2) the mappings Γa : R → R and Γ′a : R → R, where Γa(x) = x + a and Γ′a(x) = a + x, for x ∈R, are M∗-irresolute mappings of the topological space R into itself; (3) 〈S.T〉R ⊇〈S〉R.〈T〉R. 6 Int. J. Anal. Appl. (2023), 21:84 Proof. The proof of statements (1) and (2) directly follows from the definition of an M∗-irresolute topological ring R (see Definition 3.1). Statement (3) is proved in the same manner as statement (3) of Theorem 3.1. � Theorem 3.3. Let S be an M∗-open set in an M∗-irresolute topological ring R. Then S.a(respectively a.S) is M∗-open set in R for every a ∈I(R). Proof. Let x ∈ S.a be an arbitrary element. Then x = s.a for some s ∈ S. Now s = (s.a).a−1 = x.a−1 = Σa−1 (x). By Theorem 3.2(1), Σa−1 : R→R is M∗-irresolute. Thus for the M∗-open set S containing s = ∑ a−1 (x), there exists an M ∗-open set Ux in R containing x such that Σa−1 (Ux ) ⊆ S. This results in Ux.a−1 ⊆ S and hence Ux ⊆ S.a. Thus, S.a is an M∗-open set in R. � Corollary 3.1. Let S be an M∗-open set in an M∗-irresolute topological ring R. Then S+a(respectively a + S) is M∗-open set in R for every a ∈R. Remark 3.4. Theorem 3.3 and Corollary 3.1 do not hold for the choice of openness of S instead of M∗-openness of S. For instance, letR = (Z4,⊕4,�4) with τ = {∅,{0},{1},{0, 1},{1, 3},{0, 1, 3},{0, 2},{0, 1, 2},Z4}. Then, by Example 3.1, R is an M∗-irresolute topological ring. Now, consider S = {0, 1, 2} ∈ τ and 3 ∈R. Then S �4 3 = {0, 2, 3} /∈ τM∗. Similarly, S ⊕4 3 = {0, 1, 3} /∈ τM∗. Theorem 3.4. Let H be an M∗-closed set in an M∗-irresolute topological ring R. Then H.a (respec- tively a.H) is M∗-closed in R for every a ∈I(R). Proof. Let x ∈ 〈H.a〉R and U be an M∗-open set in R containing y, where y = x.a−1. Then, by Definition 3.1, there exist M∗-open sets U1 and U2 in R containing x and a−1 respectively such that U1.U2 ⊆ U. Since x ∈ 〈H.a〉R, H.a∩U1 6= ∅. Consider r ∈ H.a∩U1, then r.a−1 ∈ H∩(U1.U2) ⊆ H∩U. Consequently, H ∩ U 6= ∅, and, hence y ∈ 〈H〉R. Further, since H is M∗-closed, we have y ∈ H. Therefore x ∈ H.a. Thus 〈H.a〉R ⊆ H.a and since H.a ⊆ 〈H.a〉R is obvious. Hence H.a is M∗- closed. � Corollary 3.2. Let H be an M∗-closed set in an M∗-irresolute topological ring R. Then H + a (respectively a + H) is M∗-closed in R for every a ∈R. Remark 3.5. Theorem 3.4 and Corollary 3.2 do not hold for the choice of closedness of H instead of M∗-closedness of H. For example, let R = (Z3,⊕3,�3) with τ = {∅,{0, 1},Z3}. Then, by Example 3.2, R is an M∗- irresolute topological ring. Now, consider a closed set H = {2} in R and 2 ∈ R. Then H �3 2 = H ⊕3 2 = {1} /∈ M∗-C(R). Corollary 3.3. Let S be an M∗-open set in an M∗-irresolute topological ring R. Then −S ∈ τM∗. Int. J. Anal. Appl. (2023), 21:84 7 Theorem 3.5. Let R be an M∗-irresolute topological ring with unity and a ∈ I(R). Let E be an M∗-irresolute topological R-module, then (1) mapping Ωa : E → E (see Theorem 3.1(1)) is M∗-homeomorphism of the topological space E to itself; (2) mappings Σa : R → R and Σ′a : R → R (see Theorem 3.2(1)) are M∗-homeomorphism of the topological space R into itself. Proof. It is clearly evident that mappings Ωa, Σa and Σ′a are bijective mappings. Now, consider an M∗-open subset U of E, and x ∈ Ωa(U). Then x = Ωa(u) = a.u for some u ∈ U. From u = a−1.x and Definition 3.2, follows the existence of an M∗-open set U1 in E containing x such that a−1.U1 ⊆ U. Then, U1 ⊆ a.U = Ωa(U). This shows that Ωa(U) is an M∗-neighborhood of each of its points and, hence, Ωa(U) is an M∗-open subset of E. Thus, Ωa is pre-M∗-open mapping. Next, consider an M∗-open set U in R. Then Σa(U) = U.a. By Theorem 3.3, U.a is an M∗-open set in R. Hence, Σa is an pre-M∗-open mapping. In a similar manner, it can be proved that Σ′a is also an pre-M∗-open mapping. In view of the fact that mappings Ωa, Σa and Σ′a are M ∗-irresolute (see Theorem 3.1 and Theorem 3.2), the mappings Ωa, Σa and Σ′a are M ∗-homeomorphism. � Corollary 3.4. Let R be an M∗-irresolute topological ring and a ∈R. Then, the following statements are true: (1) the mappings Γa : R → R and Γ′a : R → R (see Theorem 3.2) are M∗-homeomorhisms of the topological space R into itself. (2) the mapping Γ : R → R, where Γ(x) = −x, are M∗-homeomorphisms of the topological space R into itself. Theorem 3.6. For any subset S of an M∗-irresolute topological ring R, 〈a.S〉R = a.〈S〉R for every a ∈I(R). Proof. Consider x ∈ 〈a.S〉R and y = a−1.x. Let U be an M∗-open set in R containing y. Then, there exist M∗-open sets U1 and U2 in R containing a−1 and x respectively such that U1.U2 ⊆ U. Since x ∈ 〈a.S〉R, a.S∩U2 6= ∅. Suppose r ∈ a.S ∩U2, then a−1.r ∈ S∩(U1.U2) ⊆ S∩U. Thus each M∗- open set containing y intersects S. Hence, y ∈ 〈S〉R and so x ∈ a.〈S〉R. Conversely, let x ∈ a.〈S〉R, then x = a.y for some y ∈ 〈S〉R. Consider an M∗-open set U in R containing x. By hypothesis, there exist M∗-open sets U1 and U2 in R containing a and y respectively such that U1.U2 ⊆ U. As y ∈ 〈S〉R, there exists some element r ∈ S ∩ U2. Then a.r ∈ (a.S) ∩ (U1.U2) ⊆ (a.S) ∩ U. Thus, (a.S) ∩ U is non empty which implies each M∗-open set containing x intersects a.S. Therefore, x ∈ 〈a.S〉R. Hence 〈a.S〉R = a.〈S〉R. � Corollary 3.5. For any subset S of an M∗-irresolute topological ring R, 〈a + S〉R = a + 〈S〉R for every a ∈R. 8 Int. J. Anal. Appl. (2023), 21:84 Lemma 3.1. Let (A,τ) and (A′,τ′) be topological spaces, let g : A → A′ be an M∗-homeomorphism, and let U ⊆ A. Then g(〈U〉A) = 〈g(U)〉A′. Theorem 3.7. Let S and T be subsets of an M∗-irresolute topological ring R. Then the following statements are true: (1) 〈S + T〉R ⊇〈S〉R + 〈T〉R; (2) 〈−S〉R = −〈S〉R; (3) 〈S −T〉R ⊇〈S〉R −〈T〉R. Proof. (1) Let x ∈ 〈S〉R + 〈T〉R and let U be an M∗-open set in R containing x. Then x = s + t, where s ∈ 〈S〉R and t ∈ 〈T〉R, and, hence, there exist M∗-open sets U1 and U2 in R containing s and t respectively, such that U1 + U2 ⊆ U. By virtue of the fact that U1 ∩S 6= ∅ and U2 ∩T 6= ∅, elements s1 ∈ U1 ∩S and t1 ∈ U2 ∩B can be found. Then, s1 + t1 ∈ S + T and s1 + t1 ∈ U1 + U2 ⊆ U, that is (S + T ) ∩U 6= ∅. Consequently, 〈S + T〉R ⊇〈S〉R + 〈T〉R. (2) Since the mapping x 7→−x is an M∗-homeomorphism of the topological space R onto itself (see Corollary 3.4), and in view of Lemma 3.1, 〈−S〉R = −〈S〉R is valid. (3) Inclusion 〈S −T〉R ⊇〈S〉R −〈T〉R results from (1) and (2). � 4. Characterizations of M∗-irresolute topological rings Theorem 4.1. Let R be an M∗-irresolute topological ring with unity, a ∈I(R) and x ∈R. Then the following statements are equivalent: (1) V is an M∗-neighborhood of element x of R; (2) V.a is an M∗-neighborhood of element x.a of R; (3) a.V is an M∗-neighborhood of element a.x of R. Proof. Consider the M∗-homeomorphism Σa : R → R (see Theorem 3.5). Since x.a = Σa(x) and V.a = Σa(V ), then (1) ⇒ (2). The mapping ψa : R → R, where ψa(y) = a.(y.a−1) for y ∈ R, is the composition of the M∗-homeomorphisms Σa−1 : R → R and Σ′a : R → R (see Theorem 3.5), hence, it is M∗- homeomorphism. Since ψa(x.a) = a.x and ψa(V.a) = a.V , then (2) ⇒ (3) By taking into consideration the M∗-homeomorphism Σ′ a−1 : R→R, equalities Σ′ a−1 (a.x) = x and Σ′ a−1 (a.V ) = V are obtained. Then from Theorem 3.5, it follows that V is an M∗-neighborhood of the element x. Thus, (3) ⇒ (1). � Corollary 4.1. Let R be an M∗-irresolute topological ring with unity and a ∈I(R). Then the following statements are equivalent: (1) V is an M∗-neighborhood of element 0 of R; (2) V.a is an M∗-neighborhood of element 0 of R; (3) a.V is an M∗-neighborhood of element 0 of R. Int. J. Anal. Appl. (2023), 21:84 9 Corollary 4.2. Let R be an M∗-irresolute topological ring and a ∈ R. Then a subset V ⊆ R is an M∗-neighborhood of a if and only if V −a is an M∗-neighborhood of 0. Proof. Consider an M∗-neighborhood V of a. Then there exists V ′ ∈ τM∗ such that a ∈ V ′ ⊆ V . Since the mapping Γ−a : R → R is an M∗-homeomorphism (see Corollary 3.4(1)), then Γ−a(V ′) = V ′ −a is an M∗-open set containing zero. Furthermore, clearly V ′ − a ⊆ V − a. Hence V − a is an M∗- neighborhood of zero. By virtue of the fact that the mapping Γa : R→R, where Γa(x) = x +a, is an M∗-homeomorphism (see Corollary 3.4(1)), the converse can be proved similarly. � Definition 4.1. A collection Ur of subsets of an M∗-irresolute topological ring R is called a basis of M∗-neighborhoods of r ∈R if any subset of Ur is an M∗-neighborhood of r and any M∗-neighborhood of the element r contains some subset from Ur. Theorem 4.2. Let U0 be a basis of M∗-neighborhoods of zero of an M∗-irresolute topological ring R. Then the following conditions are satisfied: (N1) 0 ∈ ⋂ V∈U0 V ; (N2) for any subsets U and W from U0 there exists a subset V ∈ U0 such that V ⊆ U ∩W; (N3) for any subset V ∈ U0 there exists a subset W ∈ U0 such that W + W ⊆ V ; (N4) for any subset V ∈ U0 there exists a subset W ∈ U0 such that −W ⊆ V ; (N5) for any subset V ∈ U0 there exists a subset W ∈ U0 such that W.W ⊆ V ; (N6) for any subset V ∈ U0 and any element a ∈ R there exists a subset W ∈ U0 such that a.W ⊆ V and W.a ⊆ V . Besides, if a ∈R, then Ua = {a + U | U ∈ U0} is a basis of M∗-neighborhoods of the element a. Proof. The definition of the basis of M∗-neighborhoods of an element in a topological space results in the fulfillment of conditions (N1) and (N2). Consider an M∗-neighborhood V of 0 = 0 + 0. Since R is an M∗-irresolute topological ring, then there are M∗-open sets V1 and V2 containing zero such that V1 + V2 ⊆ V . Suppose that W = V1 ∩V2, then W is M∗-neighborhood of zero and W + W ⊆ V1 + V2 ⊆ V . In a similar way, the fulfillment of conditions (N4)-(N6) results from the fact that U0 is a basis of M∗-neighborhoods of zero in an M∗-irresolute topological ring R, from conditions (C2) and (C3) (see Definition 3.1) with regard to −0 = 0, 0.0 = 0 and 0.a = 0 for any a ∈R. If a ∈ R, then the mapping Γ′a : R → R is an M∗-homeomorphism in view of Corollary 3.4, and, hence Ua = {a + U | U ∈ U0} = {Γ′a(U) | U ∈ U0} is a basis of M∗-neighborhoods of the element a. � Theorem 4.3. Let R be an M∗-irresolute topological ring and U0 be a basis of M∗-neighborhoods of zero of an M∗-irresolute topological R-module E. Then conditions (N1) to (N4) of Theorem 4.2, are satisfied together with the following conditions: 10 Int. J. Anal. Appl. (2023), 21:84 (N5′) for any subset V ∈ U0 there exist a subset W ∈ U0 and an M∗-neighborhood U of zero in R such that U.W ⊆ V ; (N6′) for any subset V ∈ U0 and any element r ∈ R there exists a subset W ∈ U0 such that r.W ⊆ V ; (N7′) for any subset V ∈ U0 and any element a ∈ E there exists an M∗-neighborhood U of zero in R such that U.a ⊆ V . Proof. To prove these conditions, it is necessary to use Theorem 4.2, condition (M3) (see Definition 3.2), and to take account of 0.a = r.0 = 0 for any r ∈R and a ∈E. � Theorem 4.4. Let S be a subset of an M∗-irresolute topological ring R and U0 be a basis of M∗- neighborhoods of zero. Then 〈S〉R = ⋂ U∈U0 (S + U). Proof. Let x ∈ 〈S〉R and U ∈ U0. Thus, by condition N3 (see Theorem 4.2), there exists W ∈ U0 such that −W ⊆ U. Since x ∈ 〈S〉R, then (x + W ) ∩ S 6= ∅. This gives x ∈ S − W ⊆ S + U. Therefore, 〈S〉R ⊆ S + U, and, hence 〈S〉R ⊆ ⋂ U∈U0 (S + U). Now, let y ∈ ⋂ U∈U0 (S + U) and let U1 be an M∗-neighborhood of zero in R. Let’s choose an M∗- neighborhood U ∈ U0 of zero such that−U ⊆ U1. Since y ∈ S + U, then S∩(y+U1) ⊇ S∩(y−U) 6= ∅. Thus ⋂ U∈U0 (S + U) ⊆〈S〉R. � Corollary 4.3. Let U0 be a basis of M∗-neighborhoods of zero of an M∗-irresolute topological ring R. Then ⋂ U∈U0 U is an M∗-closed set. Proof. By Theorem 4.4, ⋂ U∈U0 U is an M∗-closure of a subset {0} in R, and hence is M∗-closed set in R. � Corollary 4.4. Let U and W be M∗-neighborhoods of zero of an M∗-irresolute topological ring R such that W + W ⊆ U. Then 〈W〉R ⊆ U. Proof. It is obvious that the family U0 of all M∗-neighborhoods of zero in R is a basis of M∗- neighborhoods of zero of R. By Definition 4.1, for M∗-neighborhood W of zero, there exists V ∈ U0 such that V ⊆ W. In the view of Theorem 4.4, 〈W〉R = ⋂ V∈U0 (W + V ) ⊆ W + V ⊆ W + W ⊆ U. Thus, 〈W〉R ⊆ U. � Corollary 4.5. LetRbe an M∗-irresolute topological ring, E be an M∗-irresolute topologicalR-module, N ⊆ E and let U0(E) be a basis of M∗-neighborhoods of zero in E. Then following statements are true: Int. J. Anal. Appl. (2023), 21:84 11 (1) 〈N〉E = ⋂ U∈U0(E) (N + U). (2) ⋂ U∈U0(E) U is M∗-closed set in E. Theorem 4.5. Let R be an M∗-irresolute topological ring with unity, and let S be an M∗-compact subset in R. Then the following assertions hold: (1) a.S is M∗-compact for each a ∈I(R); (2) a + S is M∗-compact for each a ∈R; Proof. Let Λ be an indexing set and let {Uβ : β ∈ Λ} be an M∗-open cover of a.S. Then, a.S ⊆ ⋃ β∈Λ Uβ implies that S ⊆ ⋃ β∈Λ (a−1.Uβ). Since each Uβ is M∗-open subset of an M∗-irresolute topological ring R, then by Theorem 3.3, a−1.Uβ ∈ M∗-O(R) for each β ∈ Λ. Further, since S is M∗-compact, therefore, S ⊆ ⋃ β∈Λ′ (a−1.Uβ) for some finite subset Λ′ ⊆ Λ. Consequently, a.S ⊆ ⋃ β∈Λ′ Uβ. Hence a.S is M∗-compact. Analogously, the second part of the theorem is proved. � Definition 4.2. Let R and S be an M∗-irresolute topological rings. A mapping Ω : R → S is said to be an M∗-irresolute (pre-M∗-open) homomorphism if Ω is a homomorphism of rings and an M∗- irresolute (pre-M∗-open) mapping of the topological spaces. A homomorphism of rings, which is at the same time M∗-irresolute and pre-M∗-open, is called an M∗-topological homomorphism. Proposition 4.1. LetRandS be an M∗-irresolute topological rings, and Ω : R→S be a homomorphic mapping of R to S. Then (1) Ω is an M∗-irresolute homomorphism if and only if Ω−1(V ) is an M∗-neighborhood of zero in R for any M∗-neighborhood V of zero in S; (2) Ω is a pre-M∗-open homomorphism if and only if Ω(W ) is an M∗-neighborhood of zero in S for any M∗-neighborhood W of zero in R; (3) Ω is a M∗-topological homomorphism if and only if for any M∗-neighborhoods U and U1 of zero in R and S, correspondingly, Ω(U) and Ω−1(U1) are M∗-neighborhoods of zero in S and R respectively. Proof. (1) Let V be an M∗-neighborhood of zero in S. Then, there exists M∗-open set V ′ in S containing zero such that 0 ∈ V ′ ⊆ V . Since Ω is an M∗-irresolute homomorphism, then it is, in particular, M∗-irresolute at 0 ∈ R. Further since Ω(0) = 0, then by Definition 2.5, there exists M∗-open set U in R containing zero such that 0 ∈ Ω(U) ⊆ V ′. Consequently, 0 ∈ U ⊆ Ω−1(V ). This proves that Ω−1(V ) is an M∗-neighborhood of zero in R. Now let a ∈R and U′ be an M∗-neighborhood of Ω(a) in S. Then, due to Corollary 4.2, U′−Ω(a) is an M∗-neighborhood of zero in S. Therefore, W = Ω−1(U′−Ω(a)) is an M∗-neighborhood of zero in R. Consequently, W + a is an M∗-neighborhood of a ∈R, besides, 12 Int. J. Anal. Appl. (2023), 21:84 Ω(W + a) = Ω(W ) + Ω(a) = Ω ( Ω−1(U′ − Ω(a)) ) + Ω(a) ⊆ U′ − Ω(a) + Ω(a) ⊆ U′. Therefore, the mapping Ω : R→S is M∗-irresolute. (2) Let W be an M∗-neighborhood of zero in R. Then, there exists an M∗-open set U in R contaning zero such that 0 ∈ U ⊆ W. Since Ω is pre-M∗-open mapping, then Ω(U) is an M∗-open set in S. Besides, 0 = Ω(0) ∈ Ω(U) ⊆ Ω(W ). Thus, Ω(W ) is an M∗-neighborhood of zero in S. Conversely, consider an M∗-open subset V of R, and v ∈ Ω(V ). Then there exists an element u ∈ V such that v = Ω(u). Since V is M∗-neighborhood of point u in R, then V −u is M∗-neighborhood of zero in R. By hypothesis, Ω(V −u) is an M∗-neighborhood of zero in S and Ω(V −u) = Ω(V ) − Ω(u) = Ω(V ) −v. Thus, Ω(V ) = Ω(V −u) + v is an M∗-neighborhood of v in S. Hence, Ω(V ) is an M∗-neighborhood of each of its points. Therefore, Ω(V ) is an M∗-open set in S. This proves that Ω is pre-M∗-open mapping. (3) follows from statements (1), (2) and Definition 4.2. � Definition 4.3. Let R be an M∗-irresolute topological ring, E be an M∗-irresolute topological R- module. A subset K of E is called bounded if for any M∗-neighbourhood U of zero in E there exists an M∗-neighborhood U1 of zero in R such that U1.K ⊆ U. An M∗-irresolute topological R-module E is called bounded if E is bounded subset of the module E. Definition 4.4. Let R be an M∗-irresolute topological ring. A subset K ⊆R is called bounded from left(right) if for any M∗-neighborhood U of zero in R there exists an M∗-neighborhood U1 of zero in R such that U1.K ⊆ U (correspondingly, K.U1 ⊆ U). A subset K of an M∗-irresolute topological ring is called bounded, if it bounded from left and from right. Theorem 4.6. Let R be an M∗-irresolute topological ring, E and E′ be an M∗-irrresolute topological R-modules. Suppose Ω : E →E′ is an M∗-irresolute homomorphism and a subset K is bounded in E. Then the subset Ω(K) is bounded in E′. Proof. Let U be an M∗-neighborhood of zero in M∗-irresolute topological R-module E′. Then, due to Proposition 4.1, Ω−1(U) is an M∗-neighborhood of zero in M∗-irresolute topological R-module E. By boundedness of K, there exists an M∗-neighborhood U1 of zero in R such that U1.K ⊆ Ω−1(U). Then U1.Ω(K) = Ω(U1.K) ⊆ Ω(Ω−1(U)) ⊆ U, i.e Ω(K) is a bounded subset in E′. � Int. J. Anal. Appl. (2023), 21:84 13 Theorem 4.7. Let R and S be M∗-irresolute topological rings, Ω : R → S be an M∗-topological homomorphism. Let a subset K be bounded from left (bounded from right, bounded) in the ring R, then the subset Ω(K) is bounded from left (bounded from right, bounded) in the ring S. Proof. Let U be an M∗-neighborhood of zero in M∗-irresolute topological ring S. Then, due to Proposition 4.1, Ω−1(U) is an M∗-neighborhood of zero in M∗-irresolute topological ring R. Since K is bounded from left in R, then there exists an M∗-neighborhood U1 of zero in R such that U1.K ⊆ Ω−1(U). By Proposition 4.1, Ω(U1) is an M∗-neighborhood of zero in S. Then Ω(U1).Ω(K) = Ω(U1.K) ⊆ Ω(Ω−1(U)) ⊆ U, i.e the subset Ω(K) is bounded from left in S. � When the subset K of the ring R is bounded from right or bounded, the proof is analogous. Theorem 4.8. Every M∗-compact set in an M∗-irresolute topological R-module E is bounded. Proof. Consider an M∗-compact subset K of E. Let U be an M∗-neighborhood of zero in E. Then, by condition (M3) (see Definition 3.2), for any element n ∈ K there exist an M∗-open set Vn in R containing zero and an M∗-open set Wn in E containing n such that Vn.Wn ⊆ U. Since {Wn | n ∈ K} is an M∗-open cover of K, then there exist elements n1,n2, . . . ,ni ∈ K such that K ⊆ i⋃ j=1 Wnj . Then V = i⋂ j=1 Vnj is an M ∗-neighborhood of zero in R and V.K ⊆ V.( i⋃ j=1 Wnj ) ⊆ i⋃ j=1 (Vnj.Wnj ) ⊆ U. Thus, K is bounded subset of M∗-irresolute topological R-module E. � Theorem 4.9. M∗-closure of any bounded subset of an M∗-irresolute topological R-module E is bounded. Proof. Let K be any bounded subset of E and U be an M∗-neighbourhood of zero in E. Then there exists M∗-closed neighborhood U1 of zero in E such that U1 ⊆ U. Since K is bounded subset of E, then there exists an M∗-neighborhood U2 of zero in R such that U2.K ⊆ U1. Then U2.〈K〉E ⊆〈U2〉R.〈K〉E ⊆〈U2.K〉E ⊆〈U1〉E = U1 ⊆ U. Thus, 〈K〉E is a bounded set in E. � Theorem 4.10. If K is a bounded from left subset of an M∗-irresolute topological ring R and N is a bounded subset of an M∗-irresolute topological R-module E, then K.N is bounded subset of E. 14 Int. J. Anal. Appl. (2023), 21:84 Proof. Consider an M∗-neighborhood U of zero in E and M∗-neighborhood W of zero in R such that W.N ⊆ U. Since K is bounded from left subset of R, then there exists M∗-neighborhood V of zero in R such that V.K ⊆ W. Then V.(K.N) = (V.K).N ⊆ W.N ⊆ U. Thus, K.N is bounded subset of E. � Definition 4.5. Let (R, +, ·,τ) be an M∗-irresolute topological ring. A subset S of R is called a subring of an M∗-irresolute topological ring R if S is a subring of R and S is endowed with the topology τ|S = {V ∩S | V ∈ τ}, induced by the topology τ. Definition 4.6. Let E be an M∗-irresolute topological R-module. A subset N of E is called a submodule of the M∗-irresolute topological R-module E if N is submodule of R-module E and R-module N is endowed with the topology induced by the topology of R-module E. Theorem 4.11. Every Subring of an M∗-irresolute topological ring R is an M∗-irresolute topological ring. Proof. Consider a subring S of an M∗-irresolute topological ring R. Let r1, r2 ∈ S and U be an M∗-open set in S containing r1 − r2. Then U = V ∩ S, where V is M∗-open set in R containing r1 − r2. By Definition 3.1, there exist M∗-open sets U1 and U2 in R containing r1 and r2 respectively such that U1 −U2 ⊆ V . Then W1 = U1 ∩S and W2 = U2 ∩S are M∗-open sets in S containing r1 and r2 such that W1 −W2 ⊆ (U1 −U2) ∩S ⊆ V ∩S = U. Now, suppose r1, r2 ∈ S and U be an M∗-open set in S containing r1.r2. Then U = V ∩S, where V is M∗-open set in R containing r1.r2. By hypothesis, there exist M∗-open sets U1 and U2 in R containing r1 and r2 respectively such that U1.U2 ⊆ V . Then W1 = U1 ∩ S and W2 = U2 ∩ S are M∗-open sets in S containing r1 and r2 respectively, besides W1.W2 ⊆ (U1.U2) ∩S ⊆ V ∩S = U. Hence, S is an M∗-irresolute topological ring. � Remark 4.1. A Submodule N of an M∗-irresolute topological R-module E is an M∗-irresolute topo- logical R-module. Theorem 4.12. Let Q be a subset of an M∗-irresolute topological ring R, and S be a Q-stable subset of an M∗-irresolute topological R-module E, then 〈S〉E is a 〈Q〉R-stable subset. Proof. Since S is Q-stable subset of E, then Q.S ⊆ S (see Definition 2.8). From Theorem 3.1(3), 〈Q〉R.〈S〉E ⊆〈Q.S〉E ⊆〈S〉E. Thus, 〈S〉E is a 〈Q〉R-stable subset. � Int. J. Anal. Appl. (2023), 21:84 15 Proposition 4.2. Let R be an M∗-irresolute topological ring and E be an M∗-irresolute topological R-module. Let Q be a subring of R, and N be a Q-submodule of R-module E, then (1) 〈Q〉R is a subring R; (2) 〈N〉E is a 〈Q〉R-module. Proof. Let Q be a subring of an R. Thus Q−Q ⊆ Q and Q.Q ⊆ Q. By Theorem 3.7(3), 〈Q〉R − 〈Q〉R ⊆〈Q−Q〉R ⊆〈Q〉R. Since Q is a Q-stable subset of the M-irresolute topological R-module R, then, due to Theorem 4.12, 〈Q〉R is a 〈Q〉R-stable subset of the M∗-irresolute topological R-module R, that is 〈Q〉R.〈Q〉R ⊆〈Q〉R. Hence 〈Q〉R is a subring R. Since N is Q-submodule of R-module E, then by Theorem 3.7(1) 〈N〉E +〈N〉E ⊆〈N +N〉E ⊆〈N〉E. Also N is Q-stable subset of an M∗-irresolute topological R-module E, then by Theorem 4.12, 〈N〉E is a 〈Q〉R-stable subset of the M∗-irresolute topological R-module E, and since 〈Q〉R is a subring R, then 〈N〉E is a 〈Q〉R-module. � Corollary 4.6. Let Q be a subring of an M∗-irresolute topological ring R with 〈Q〉R = R and N be a Q-submodule of an M∗-irresolute topological R-module E Then, 〈N〉E is a submodule of the M∗-irresolute topological R-module E. In particular, the M∗-closure of any submodule of an M∗-irresolute topological R-module is also an M∗-irresolute topological R-module. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] V.I. Arnautov, M.I. Vodichar, A.V. Mikhalev, Introduction to The Theory of Topological Rings and Modules, Kishinev, Stiintsa, 1981. [2] N. Bourbaki, Éléments de Mathématique, Topologie Générale, Hermann, Paris, (1965). [3] M. Caldas, S. Jafari, M.M. 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