EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6261 ISSN 1307-5543 – ejpam.com Published by New York Business Global Direct Product of Complex Neutrosophic Subrings Muhammad Haris Mateen1, Sarka Hoskova-Mayerova2,∗, Kholood Alnefaie3, Florentin Smarandache4 1 School of Mathematics, Minhaj University Lahore, Pakistan 2 Department of Mathematics and Physics, University of Defence, 66210 Brno, Czech Republic 3 Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia 4 University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA Abstract. The complex neutrosophic set is a generalization of the neutrosophic set with the addition of three phase terms. The complex neutrosophic set deals with periodic data that contains uncertainty, indeterminacy, and falsity. The complex neutrosophic set has a variety of applications, such as signal processing, hospital infrastructure design, medical image denoising, segmentation distance measurement, and the game of loser, neutral, and winner. This article presents a novel concept for complex neutrosophic subrings and illustrates how these subrings can generate two other neutrosophic subrings. Additionally, we prove that the intersection of two neutrosophic subrings is a neutrosophic subring. We expand this idea to talk about the abstraction of level subsets of complex neutrosophic sets and look into the basic algebraic properties of this event. We prove that the level subset of the complex neutrosophic subring is a subring. Moreover, we demonstrate that the product of two complex neutrosophic subrings is also a complex neutrosophic subring and explore some novel consequences about the direct product of complex neutrosophic subrings. Our findings generalize and extend the existing ring theory results within a complex neutrosophic framework. 2020 Mathematics Subject Classifications: 13E15, 08A72, 03E72 Key Words and Phrases: Complex neutrosophic subring, level subsets of complex neutrosophic subrings, product of complex neutrosophic subrings 1. Introduction In the beginning, initiatives to demonstrate Fermat’s last theorem gave rise to the idea of a ring in the 1880s, commencing with Dedekind [1]. In the 1920s, Noether and Krull [2] ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6261 Email addresses: harism.math@gmail.com (M. H. Mateen), sarka.mayerova@unob.cz (S. Hoskova-Mayerova), knefaie@taibahu.edu.sa (K. Alnefaie), smarand@unm.edu (F. Smarandache) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 2 of 23 generalized and firmly established the idea of a ring after contributions from other domains, especially number theory. Modern ring theory, a relatively active mathematical field, inves- tigates rings independently. Fuzzy set theory conceptualized by Zadeh [3] as an extension of the idea of classical set theory, It is extremely important for managing uncertainty in practical applications and described as κ :→ Q such that {y : (y, κQ(y)) ∀ y, κQ(y) ∈ [01]} . As an extension of the fuzzy set, Atanassov [4] presented an intuitionistic fuzzy (IF) set and characterize as χ : T → {(x, µ(y), µ(y)) : y ∈ T} where , µ(y), µ(y) ∈ [0 1] such that 0 ≤ µ(y) + µ(y) ≤ 1. In decision-making challenges, the positive and negative member- ship functions of IF sets in contrast to classical fuzzy sets, provide that both are able to manage situations that are unclear and uncertain in physical issues. Alolaiyan [5] et al. discussed decision making problems by employing linguistic intuitionistic fuzzy set with a fuzzy Dombi weighted geometric operator. Rosenfeld [6] proposed the fuzzy subgroup in 1971 by applying the fuzzy set theory on algebra. IF subgroups and the algebraic structure of intuitionistic fuzzification introduced by Biswas [7]. Smarandache [8] was the first to introduce neutrosophy as a discipline of philosophy that investigated the origin, nature, and scope of neutralities as well as how they interacted with various ideational spectra. A belonging membership function, a not belonging mem- bership function, and an indeterminacy membership function define a neutrosophic set (NS). Agboola et al. [9] provided the idea of neutrosophic BCI/BCK algebras and dis- cussed some fundamental characteristics of neutrosophic BCI/BCK algebras. The group structure of single valued NSs was investigated by Cetkin and Aygun [10]. Additionally, they scrutinized the essential features of the neutrosophic subgroup and showcased the homomorphic image and pre-image of a neutrosophic (normal) subgroup. Song et al. [11] offered the idea of a neutrosophic distributive N -ideal in BCK-algebras and explored a number of its features. Additionally, they engaged in a discussion about the connections between a neutrosophic commutative N -ideal and a neutrosophic N -ideal. Chalapathi and Kumar [12] discussed the finite groups through graphs under the framework of NS. The idea of the neutrosophic triplet group, It includes a novel extension of the traditional group idea, was derived from the fundamental idea of the NS and the structural characteristics of the neutrosophic triplet group are examined in detail [13]. In a BCK-algebra, Bor- zooei et al. [14] developed the idea of an extended neutrosophic commutative ideal, and associated features were demonstrated. Moreover, some equivalence relations have been introduced, and several features of the family of all commutative modified neutrosophic ideals in BCK-algebras are investigated. Interval neutrosophic subalgebra was introduced by Jun et al. [15]. In BCK/BCI-algebra, their characteristics and relationships are stud- ied. Additionally, we introduce the concept of an interval neutrosophic length and review its associated features. The idea of a neutrosophic positive implicative N -ideal in BCK- algebras was suggested by Jun et al. [16], and numerous features were studied. Bender et al. [17] investigated the complex anti-fuzzy subgroups and proved some important results. Gulzar et al. [18] discussed the Q-fuzzy subrings under the framework of complex fuzzy (CF) sets. Alghazzawi et al. [19] purposed the optimal solution for energy crises by employing the interval-valued intuitionistic fuzzy sets. Alolaiyan et al. [20] studied the algebraic M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 3 of 23 structure of bipolar fuzzy subrings. Additionally, a detailed discussion of the algebraic attributes took place. Altassan et al. [21] discussed the algebraic product of fuzzy sub- rings. Furthermore, the fundamental theorems of the fuzzy isomorphism subring extend to algebraic products. Dilshad et al. [22] introduced the novel idea of q-rung orthopair fuzzy (ROF) subrings. Under the influence of ring theory, they also established various algebraic operations, ideals, homomorphic images, and pre-images. Mateen et al. initiated the novel algebraic structure of complex Pythagorean fuzzy subfield. They also explored the direct product and homomorphism within the context of a complex Pythagorean fuzzy set. Bal et al. [23] defined the neutrosophic extended triplet subgroup, neutrosophic ker- nel, neutrosophic inverse-image, and neutrosophic image in this study using the idea of a neutrosophic extended triplet. Smarandache et al. [24] examined the neutrosophic triplet G-module and described its characteristics. Also, the definitions of neutrosophic triplet G-modules that are reducible, irreducible, and completely reducible were given, along with an analysis of the connections between these structures. Ali et al. established a neutrosophic triplet subring and neutro- sophic triplet subfield, as well as some of its fundamental characteristics. In a neutrosophic group, Abobala et al. [25] defined certain novel substructures (AH-substructures). It also covers some basic AH-subgroup characteristics, AH-normality, AH-homomorphisms, AH-quotients, and AH-direct products. Kandasamy et al. [26] discussed the neutrosophic triplets in neutrosophic rings < Z ∪ I > < Q ∪ I > or < R ∪ I >. Furthermore, the study revealed the existence of three distinct types of neutrosophic triplets in these neutrosophic subrings, all of which generate torsion-free component-wise product abelian groups. The idea of a generalized neutrosophic extended triplet group is introduced by Ma et al. [27], and some of its features are addressed. Researchers have demonstrated that the generalized neutrosophic extended triplet group and the weak commutative generalized neutrosophic extended triplet group are equivalent to the quasi-completely regular semigroup and the quasi-clifford semigroup, respectively. Numerous characteristics of implicative neutrosophic quadruple BCK-algebras were investigated by Muhiuddin et al. [28], and developed criteria for the neutrosophic quadru- ple set to be a neutrosophic quadruple BCI-algebra. Kandasamy et al. [26] examined the semi-idempotents in neutrosophic subrings and studied the vital characteristics in de- tails. Bashir et al. [29] introduced the subsemirings, ideals, generalized bi-ideals, and quasi-ideals under the framework of a m-polar fuzzy set in semirings. Bashir et al. [30] purposed the ternary multiplication to extend the roughness of a fuzzy set in three dimen- sions. A number of vital characteristics were discussed by using the idea of set-valued and strong set-valued homomorphism. The notion of CF sets and their fundamental algebraic operations were investigated in [31, 32], extending the range of the belonging function from real numbers to complex numbers with the unit disc. Due to the CF set only considering the degree of belonging and giving no consideration to the data entities that are not members, which are equally important in the process of making decisions on system evaluation. However, it is usually difficult to measure the true value of a truth estimate by the exact value of a fuzzy set in real life. In some circumstances, it could be simpler to express the ambiguity and vagueness M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 4 of 23 that characterize real-world situations using two-dimensional data rather than one. The complex intuitionistic fuzzy (CIF) sets and their set operations were presented by Alkouri and Salleh [33, 34]. Also, it indicated the unpredictability of complex-valued functions in a variety of physical measurements. As an example, complex intensity, impedance and wave function in the fields of quantum physics and electronics. The following are the motivations for the new work. (i) The concept of a CIF subgroup and CIF level subsets were introduced by Gulzar et al. [35]. Additionally, the Cartesian product of two CIF subgroups was introduced, along with the homomorphic image and inverse image of the CIF subgroup under group homomorphism. (ii) Gulzar et al. [36] introduced the idea of the direct product of two CIF subrings, demonstrated that it is also a CIF subring, and discussed about its different algebraic characteristics. Hameed et al. [37] discussed the (α, β, γ) neutrosophic submodules and some fundamental algebraic properties were investigated. (iii) Elrawy and Abdalla [38] defined the algebraic structure based on single-valued NSs and proposed a novel method for constructing the neutrosophic subring and ideal by combining NSs with the classical ring. (iv) Ali and Smarandache established innovative complex neutrosophic sets (CNS), which extend the range of components in the complex plane from the unit interval to the unit disc. Amplitude and phase values are assigned to each of its components. Furthermore, CNSs have been used in the fields of science and engineering. (v) Gulistana et al. [39] initiated the idea of complex neutrosophic subgroups and de- fined the term alpha-cut of CNS. The cartesian product of complex neutrosophic subgroups is also defined. Rahoumah et al. [40] introduced the complex neutro- sophic soft subgroups, and the fundamental results related to this phenomenon were discussed. (vi) The complex neutrosophic set is the generalization of existing theorizes i.e., complex fuzzy sets and complex intuitionistic fuzzy sets. The concept of complex neutrosophic set is not yet applied to submodules. Our proposed model is given in Figure 1. In this paper, we initiate the work on complex neutrosophic subrings (CNSRs). The paper is shaped as follow: The concept of CNS is defined in Section 2. Some important algebraic properties of this abstraction are mentioned in this section. In Section 3 the novel concept of CNSR is present together with their fundamental results. We show that the intersection of two CNSRs is CNSR. Additionally, we depict level subset of complex neutrosophic(CN) subset and prove that the level subsets of CNSR is CNR. The 4th Section reveals the idea about direct product of CNSRs and explore algebraic characteristics. We shown that direct product of CNSRs is CNSR. M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 5 of 23 Figure 1: Flowchart of Proposed Model 2. Preliminaries The following portion recalls some essential concepts of CNS and CNSR which are necessary for our further discussion. Definition 1. [8] NS W of universal set P is of the form W = {< c, pW (c), qW (c), rW (c) >: c ∈ P}, where pW (c), qW (c) and rW (c) give the degree of accuracy, level of indeterminacy and degree of falsehood of c from unit interval, respectively such that 0 ≤ pW (c) + qW (c) + rW (c) ≤ 3, for any c ∈ P . Definition 2. [38] A NS W of a ring R is known a NSR of a R, if these conditions are valid: (i) pW (m− d) ≥ min{pW (m), pW (d)}, for all m,d ∈ R. (ii) pW (md) ≥ min{pW (m), pW (d)}, (iii) qW (md) ≤ max{qW (m), qW (d)}, (iv) qW (m− d) ≤ max{qW (m), qW (d)}, (v) rW (m− d) ≤ max{rW (m), rW (d)}, M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 6 of 23 Table 1: Comparison of complex neutrosophic sets to the existing approaches Sets Domain Co-domain Truth Falsity Indeterminacy Truth Falsity Indeterminacy with periodicity with periodicity with periodicity Fuzzy sets Universel set Unit interval X × × × × × Intuitionistic fuzzy sets Universel set unit interval X X × × × × Neutrosophic sets Universel set Unit interval X X X × × × Complex fuzzy sets Universel set Unit Disc X × × X × × Complex intuitionistic fuzzy sets Universel set Unit Disc X X × X X × Complex Neutrosophic fuzzy sets Universel set Unit Disc X X X X X X (vi) rW (md) ≤ max{rW (m), rW (d)}. Example 1. Consider the ring R = Z6 = {0, 1, 2, 3, 4, 5} under addition and multiplication modulo 6. Define a neutrosophic subring N of R as follows: pW (x) =  1, if x = 0, 0.88, if x ∈ {2, 4}, 0, otherwise. qW (x) =  0, if x = 0, 0.11, if x ∈ {2, 4}, 0.33, otherwise. rW (x) =  0, if x = 0, 0.11, if x ∈ {2, 4}, 0.77, otherwise. Clearly, N is NSR of subring R. Theorem 1. [38] Intersection of two NSRs of ring R is NSR. Definition 3. [31, 32] A CNS W of universal set P is an object of the form W = {< f,TW(f), IW(f),FW(f) >: f ∈ P}, where the degree of truth TW(f) = pW(f)eiθW(f), and is expressed as TW : P → {τ ∈ C : |τ | ≤ 1}, degree of indeterminacy IW(f) = qW(f)eiφW(f) is expressed as IW : P → {τ ∈ C : |τ | ≤ 1} and degree of falsity FW(f) = rW(f)eiωW(f) and is defined as FW : P → {τ ∈ C : |τ | ≤ 1}, where |TW(f) + IW(f) + FW(f)| ≤ 3 and C is the set of complex numbers. the degree of accuracy, level of indeterminacy and degree of falsehood receive all complex valued grade from within in the unit circle of complex plane, respectively. where i = √ −1 , pW(f), qW(f), rW(f) ∈ [0, 1], and θW(f), φW(f), ωW(f) ∈ [0, 2π] are real valued such that 0− ≤ pW(f) + qW(f) + rW(f) ≤ 3+ and 0 ≤ θW(f) + φW(f) + ωW(f) ≤ 6π. For convenience we shall use TW(f) = pW(f)eiθW(f), TX(f) = pX(f)e iθX(f) as degree of truth, IW(f) = qW(f)eiφW(f), IX(f) = qX(f)e iφX(f) as degree of indeterminacy and FW(f) = rW(f)eiωW(f), FX(f) = rX(f)e iω X (f) as degree of falsity of CNSs W and X. M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 7 of 23 Definition 4. [41] Assume that W and X be two CNSs of set P. Then the intersection of CNSs W and X is elaborated as: W ∩ X = {< (p),TW∩X(p), IW∩X(p),FW∩X(p) >}. Where TW∩X(p) = pW∩X(p)e iθW∩X(p) = min{pW(p), pX(p)}eimin{θW(p),θX(p)}, IW∩X(p) = qW∩X(p)e iφW∩X(p) = max{qW(p), qX(p)}eimax{φW(p),φX(p)}, FW∩X(p) = rW∩X(p)e iωW∩X(p) = max{rW(p), rX(p)}eimax{ωW(p),ωX(p)}. Definition 5. [41] Let W and X be two CNSs of set P . Then the union of CNSs W and X is described as: W ∩ X = {< (p),TW∩X(p), IW∩X(p),FW∩X(p) >}. Where TW∩X(p) = pW∩X(p)e iθW∩X(p) = max{pW(p), pX(p)}eimax{θW(p),θX(p)}, IW∩X(p) = qW∩X(p)e iφW∩X(p) = min{qW(p), qX(p)}eimin{φW(p),φX(p)}, FW∩X(p) = rW∩X(p)e iωW∩X(p) = min{rW (p), rX(p)}eimin{ωW(p),ωX(p)}. 3. Properties of Complex Neutrosophic Subrings The investigation of CNSRs and level subsets of CNSRs is the focus of this section. We investigate that a CNSRs produces two neutrosophic subrings and intersection of two CNSRs is CNSR. We describe the level-subset of CNS and prove that level-subset of CNSR form subring of ring. Definition 6. Let W = {< f, θW (f), φW (f), ωW (f), >: f ∈ K} be a NS, where k is the ring. Then the π-NS Wπ is described as Wπ = {< f, θWπ(f), φWπ(f), ωWπ(f) >: f ∈ K}, where the function θWπ(f) = 2πθW (f), φWπ(f) = 2πφW (f) and ωWπ(f) = 2πωW (f) indicate the measure of association, measure of indeterminacy and measure of non- association of an element f of K, consequently. and fulfil the consequent criteria 0 ≤ θWπ(f) + φWπ(f) + ωWπ(f) ≤ 6π. Definition 7. A π-NS Wπ of ring K is known as π-neutrosophic subring of K, ∀ p, u ∈ K if (i) θWπ(p− u) ≥ min{θWπ(p), θWπ(u)} , (ii) θWπ(pu) ≥ min{θWπ(p), θWπ(u)} , (iii) φWπ(p− u) ≤ max{φWπ(p), φWπ(u)} , (iv) φWπ(pu) ≤ max{φWπ(p), φWπ(u)} , (v) ωWπ(p− u) ≤ max{ωWπ(p), ωWπ(u)} , (vi) ωWπ(pu) ≤ max{ωWπ(p), ωWπ(u)} . Definition 8. Let W and X be two CNSs of K. Then M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 8 of 23 (i) A CNS W is homogeneous CNS, if for all p, a ∈ K, we have a pW (p) ≤ pW (a) if and only if θW (p) ≤ θW (a), b qW (p) ≤ qW (a) if and only if φW (p) ≤ φW (a), c rW (p) ≥ rW (a) if and only if ωW (p) ≥ ωW (a). (ii) A CNS W is homogeneous CNS with X, if for all p ∈ K, we have a pW (p) ≤ pX(p) iff θW (p) ≤ θX(p), b qW (p) ≤ qX(p) iff φW (p) ≤ φX(p), c rW (p) ≥ rX(p) iff ωW (p) ≥ ωX(p). In this article, we shall take CNS as homogeneous CNS. Definition 9. A CNS W = {< (x,TW (p), IW (p),FW (p)) >: p ∈ K} of ring K is called a CNSR, ∀ p, u ∈ K if (i) TW (p− u) ≥ min{TW (p),TW (u)} , (ii) TW (pu) ≥ min{TW (p),TW (u)} , (iii) IW (p− u) ≤ max{IW (p), IW (u)} , (iv) IW (pu) ≤ max{IW (p), IW (u)} , (v) FW (p− u) ≤ max{FW (p),FW (u)} , (vi) FW (pu) ≤ max{FW (p),FW (u)} , Another way to describe the definition CNSR is given as; (i) pW (p− u)eiθW (p−u) ≥ min{pW (p), pW (u)}eimin{θW (p),θW (u)}, (ii) pW (pu)eiθW (pu) ≥ min{pW (p), pW (u)}eimin{θW (p),θW (u)}, (iii) qW (p− u)eiφW (p−u) ≤ max{qW (p), qW (u)}eimax{φW (p),φW (u)}, (iv) qW (pu)eiφW (pu) ≤ max{qW (p), qW (u)}eimax{φW (p),φW (u)}, (v) rW (p− u)eiωW (p−u) ≤ max{rW (p), rW (u)} eimax{ωW (p),ωW (u)} (vi) rW (pu)eiωW (pu) ≤ max{rW (p), rW (u)} eimax{ωW (p),ωW (u)} for all p, u ∈ K. In the following theorem, we prove that a CNSR produce two neutrosophic subrings, namely neutrosophic subring and π-neutrosophic subring(π- NSR). Theorem 2. Let W be a CNS of ring K. Then W is a CNSR of K if and only if (i) The fuzzy set W = {< k, pW (k), qW (k), rW (k) >: k ∈ K, pW (k), qW (k), rW (k) ∈ [0, 1] and 0 ≤ pW (k) + qW (k) + rW (k)+ ≤ 3} is a NSR . M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 9 of 23 (ii) The π-fuzzy set W = {< k, θW (k), φW (k), ωW (k) >: k ∈ K, θW (k), φW (k), ωW (k) ∈ [0, 2π]} is a π- NSR. Proof. Assume that W is a CNSR and k, l ∈ M . Then we know that, pW (k − l)eiθW (k−l) = TW (k − l) ≥ min{TW (k),TW (l)} = min{pW (k)eiθW (k), pW (l)eiθW (l)} = min{pW (k), pW (l)}eimin{θW (k),θW (l)}. As W is homogeneous, so pW (k − l) ≥ min{pW (k), pW (l)} and θW (k − l) ≥ min{θW (k), θW (l)}. pW (kl)eiωW (kl) = TW (kl) ≥ min{TW (k),TW (l)} = min{pW (k)eiθW (k), pW (l)eiθW (l)} = min{pW (k), pW (l)}eimin{θW (k),θW (l)}. As W is homogeneous, thus pW (kl) ≥ min{pW (k), pW (l)} and θW (kl) ≥ min{θW (k), θW (l)}. Suppose that W is a CNSR and k, l ∈ H. Then we have, qW (k − l)eiφW (k−l) = IW (k − l) ≤ max{IW (k), IW (l)} = max{qW (k)eiφW (k), qW (l)eiφW (l)} = max{qW (k), qW (l)}eimax{φW (k),φW (l)}. As W is homogeneous, so qW (k − l) ≤ max{qW (k), qW (l)} and φW (k − l) ≤ max{φW (k), φW (l)}. qW (kl)eiφW (kl) = IW (kl) ≤ max{IW (k), IW (l)} = max{qW (k)eiφW (k), qW (l)eiφW (l)} = max{qW (k), qW (l)}eimax{φW (k),φW (l)}. As W is homogeneous, we have qW (kl) ≤ max{qW (k), qW (l)} and φW (kl) ≤ max{φW (k), φW (l)}. rW (k − l)eiωW (k−l) = rW (k − l) ≤ max{FW (k),FW (l)} = max{rW (k)eiωW (k), rW (l)eiωW (l)} = max{rW (k), rW (l)}eimax{ωW (k),ωW (l)}. M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 10 of 23 As W is homogeneous, so rW (k − l) ≤ max{rW (k), rW (l)} and ωW (k − l) ≤ max{ωW (k), ωW (l)}. rW (kl)eiωW (kl) = FW (kl) ≤ max{FW (k),FW (l)} = max{rW (k)eiωW (k), rW (l)eiωW (l)} = max{rW (k), rW (l)}eimax{ωW (k),ωW (l)}. As W is homogeneous, so rW (kl) ≤ max{rW (k), rW (l)} and ωW (kl) ≤ max{ωW (k), ωW (l)}. Consequently, W is NSR and W is π-NSR. Conversely, assume that W and W is NSR and π-NSR, respectively. Then, we know that pW (k − l) ≥ min{TW (k), TW (l)}, pW (kl) ≥ min{pW (k), pW (l)}, qW (k − l) ≤ max{qW (k), qW (l)}, qW (kl) ≤ max{qW (k), qW (l)}, rW (k − l) ≤ max{rW (k), rW (l)}, rW (kl) ≤ max{rW (k), rW (l)} θW (k − l) ≥ min{θW (k), θW (l)}, θW (kl) ≥ min{θW (k), θW (l)}, φW (k − l) ≤ max{φW (k), φW (l)}, φW (kl) ≤ max{φW (k), φW (l)}, ωW (k − l) ≤ max{ωW (k), ωW (l)}, ωW (kl) ≤ max{ωW (k), ωW (l)}, For this, we consider TW (k − l) = pW (k − l)eiθW (k−l) ≥ min{pW (k), pW (l)}eimin{θW (k),θW (l)} = min{pW (k)eiθW (k), pW (l)eiθW (l)} = min{TW (k),TW (l)} . For this, we have TW (kl) = pW (kl)eiθW (kl) ≥ min{pW (k), pW (l)}eimin{θW (k),θW (l)} = min{pW (k)eiθW (k), pW (l)eiθW (l)} = min{TW (k),TW (l)} . For this, we get IW (k − l) = qW (k − l)eiφW (k−l) ≤ max{qW (k), qW (l)}eimax{φW (k),φW (l)} = max{qW (k)eiφW (k), qW (l)eiφW (l)} = max{IW (k), IW (l)} . IW (kl) = qW (kl)eiφW (kl) ≤ max{qW (k), qW (l)}eimax{φW (k),φW (l)} = max{qW (k)eiφW (k), qW (l)eiφW (l)} = max{IW (k), IW (l)} . M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 11 of 23 Consider, FW (k − l) = rW (k − l)eiωW (k−l) ≤ max{rW (k), rW (l)}eimax{ωW (k),ωW (l)} = max{rW (k)eiωW (k), rW (l)eiωW (l)} = max{FW (k),FW (l)} . FW (kl) = rW (kl)eiωW (kl) ≤ max{rW (k), rW (l)}eimax{ωW (k),ωW (l)} = max{rW (k)eiωW (k), rW (l)eiωW (l)} = max{FW (k),FW (l)} . Hence, W is CNSR. In the following theorem, we illustrate that the intersection of two complex neutro- sophic subrings is complex neutrosophic subring. Theorem 3. Intersection of two CNSRs of ring R is CNSR. Proof. The proof is so accomplished. Remark 1. The union of two CNSRs of ring R may not be CNSR of ring R. In the upcoming example, we describe that the union of two CNSRs of ring may not be CNSR of ring. Example 2. Take R = Z = {0,±1,±2,±3, ...} is a ring of integers. Assume that W and X are two CNSRs of ring R and described as TW(q) = { 0.3e iπ 3 if q ∈ 3Z 0 otherwise. IW(q) = { 0.4e iπ 3 if q ∈ 3Z 0 otherwise. FW(q) = { 0.2e iπ 9 if q ∈ 3Z 0.6e iπ 4 else. TX(q) = { 0.2e iπ 3 if q ∈ 2Z 0.01e iπ 8 otherwise. IX(q) = { 0.1e iπ 4 if q ∈ 2Z 0.01e iπ 8 otherwise. FX(q) = { 0.4e iπ 8 if q ∈ 2Z 0.2e iπ 9 otherwise. M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 12 of 23 It is simple to determine that W and X are two CNSRs of ring R. Using Definition 5 W ∪ X = {(q,TW∪X, IW∪X,FW∪X)}. Therefore, TW∪X(q) =  0.3e iπ 3 if q ∈ 3Z 0.2e iπ 3 if q ∈ 2Z− 3Z 0.01e iπ 8 otherwise. IW∪X(q) =  0.4e iπ 3 if q ∈ 3Z 0.1e iπ 4 if q ∈ 2Z− 3Z 0.01e iπ 4 else. FW∪X(q) =  0.2e iπ 9 if q ∈ 3Z 0.4e iπ 9 if q ∈ 2Z− 3Z 0.2e iπ 9 else. Assume s = 15 and t = 10. Then TW∪X(15) = 0.3e iπ 3 and TW∪X(10) = 0.2e iπ 3 , then TW∪X(15−10) = TW∪X(5) = 0.01e iπ 8 and min{TW∪X(15),TW∪X(10)} = min{0.3e iπ 3 , 0.2e iπ 3 } = 0.2e iπ 3 . Clearly, TW∪X(15 − 10) < min{TW∪X(15),TW∪X(10)}. This condition does not holds. Consequently, W ∪ X is not CNSRs of R. Now, we define the idea of complex neutrosophic level subset of CNS and discuss its vital results under the framework of complex neutrosophic subring. Definition 10. Let W = {< j,TW(j), IW(j),FW(j) >: j ∈ R} be a CNS of R, for all ν, η, ρ ∈ [0, 1], and ν̂, η̂, ρ̂ ∈ [0, 2π]. The level subset of CNS is described as; W(ν, η, ρ) (ν̂, η̂, ρ̂) = {j ∈ R : pW(j) ≥ ν, θW(j) ≥ ν̂, qW(j) ≥ η, φW(j) ≥ η̂, rW(j) ≤ ρ, ωW(j) ≤ ρ̂}. For η = ρ = 0 = η̂ = ρ̂, we get Wν ν̂ = {j ∈ R : pW(j) ≥ ν, θW(j) ≥ ν̂}, for ν = ρ = 0 = ν̂ = ρ̂, we get, Wη η̂ = {j ∈ R : qW(j) ≥ η, φW(j) ≥ η̂} and for ν = η = 0 = ν̂ = η̂, we get Wρ ρ̂ = {j ∈ R : rW(j) ≤ ρ, ωW(j) ≤ ρ̂}. Theorem 4. Let W be CNNSR of ring R. Then W(ν, η, ρ) (ν̂, η̂, ρ̂) is a subring of ring R, for all ν, η, ρ ∈ [0, 1], and ν̂, η̂, ρ̂ ∈ [0, 2π], where pW(j) ≥ ν, θW(j) ≥ ν̂, qW(j) ≥ η, φW(j) ≥ η̂, rW(j) ≤ ρ, ωW(j) ≤ ρ̂. Proof. We know that W(ν, η, ρ) (ν̂, η̂, ρ̂) is nonempty, as e ∈ A (ν, η, ρ) (ν̂, η̂, ρ̂). Let f, y ∈ A (ν, η, ρ) (ν̂, η̂, ρ̂) be any two elements. Then pW(j) ≥ ν, θW(j) ≥ ν̂, qW(j) ≥ η, φW(j) ≥ η̂, rW(j) ≤ ρ, ωW(j) ≤ ρ̂. Now we suppose that, pW(j− g)eiθW(j−g) = TW(j− g) ≥ min{TW(j),TW(g)} = min{pW(j)eiθW(j), pW(g)eiθW(g)} M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 13 of 23 = min{pW(j), pW(g)} eimin{θW(j),θW(g)} . As W is homogeneous, so pW(j− g) ≥ min{pW(j), pW(g)} = min{ν, ν} = ν, θW(j− g) ≥ min{θW(j), θW(g)} = min{ν̂, ν̂} = ν̂. pW(jg)eiθW(jg) = TW(jg) ≥ min{TW(j),TW(g)} = min{pW(j)eiθW(j), pW(g)eiθW(g)} = min{pW(j), pW(g)} eimin{θW(j),θW(g)} . As W is homogeneous, so pW(jg) ≥ min{pW(j), pW(g)} = min{ν, ν} = ν, θW(jg) ≥ min{θW(j), θW(g)} = min{ν̂, ν̂} = ν̂. Further, qW(j− g)eiφW(j−g) = IW(j− g) ≥ min{IW(j), IW(g)} = min{qW(j)eiφW(j), qW(g)eiφW(g)} = min{qW(j), qW(g)} eimin{φW(j),φW(g)} . As W is homogeneous, so qW(j− g) ≥ min{qW(j), qW(g)} = min{ν, ν} = ν, φW(j− g) ≥ min{φW(j), φW(g)} = min{ν̂, ν̂} = ν̂. qW(jg)eiφW(jg) = IW(jg) ≥ min{IW(j), IW(g)} = min{qW(j)eiφW(j), qW(g)eiφW(g)} = min{qW(j), qW(g)} eimin{φW(j),φW(g)} . As W is homogeneous, so qW(jg) ≥ min{qW(j), qW(g)} = min{ν, ν} = ν, φW(jg) ≥ min{φW(j), φW(g)} = min{ν̂, ν̂} = ν̂. Further, rW(j− g)eiωW(j−g) = FW(j− g) ≤ max{FW(j),FW(g)} = max{rW(j)eiωW(j), rW(g)eiωW(g)} = max{rW(j), rW(g)} eimax{ωW(j),ωW(g)} . By homogeneity, so rW(j− g) ≤ max{rW(j), rW(g)} = max{ρ, ρ} = ρ, ωW(j− g) ≤ max{ωW(j), ωW(g)} = max{ρ̂, ρ̂} = ρ̂. M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 14 of 23 rW(jg)eiωW(jg) = FW(jg) ≤ max{FW(j),FW(g)} = max{rW(j)eiωW(j), rW(g)eiωW(g)} = max{rW(j), rW(g)} eimax{ωW(j),ωW(g)} . By homogeneity, so rW(jg) ≤ max{rW(j), rW(g)} = max{ρ, ρ} = ρ, ωW(jg) ≤ max{ωW(j), ωW(g)} = max{ρ̂, ρ̂} = ρ̂. This implies that mn ∈ W(ν, η, ρ) (ν̂, η̂, ρ̂). Hence, W(ν, η, ρ) (ν̂, η̂, ρ̂) is subring. Theorem 5. Let W(α, β, γ) (α̂, β̂, γ̂) be a subring of ring R, then W is CNSR of R if pW(j) ≥ α, θW(j) ≥ α̂, qW(j) ≥ β, φW(j) ≥ β̂, rW(j) ≤ γ, ωW(j) ≤ γ̂, ∀ α, β, γ ∈ [0, 1], and α̂, β̂, γ̂ ∈ [0, 2π]. Proof. Suppose that min{pW(j), pW(g)} = α, min{θW(j), θW(g)} = α̂,min{qW(j), qW(g)} = β, min{φW(j), φW(g)} = β̂ and max{rW(j), rW(g)} = γ, max{ωW(j), ωW(g)} = γ̂. Then we have pW(j) ≥ α, qW(j) ≥ β, rW(j) ≤ γ, θW(j) ≥ α̂, φW(j) ≥ β̂, ωW(j) ≤ γ̂ and pW(g) ≥ α, qW(g) ≥ β, rW(g) ≤ γ, θW(g) ≥ α, φW(g) ≥ β, ωW(g) ≤ γ̂. This means that f ∈ W(α, β, γ) (α̂, β̂, γ̂) and n ∈ W(α, β, γ) (α̂, β̂, γ̂) . As W(α, β, γ) (α̂, β̂, γ̂) is subring, so mn ∈ W(α, β, γ) (α̂, β̂, γ̂) . Then we have pW(j− g) ≥ α and θW(j− g) ≥ α̂, qW(j− g) ≥ β and φW(j− g) ≥ β̂, rW(j− g) ≤ γ and ωW(j− g) ≤ γ̂ Implies that pW(j− g) ≥ min{pW(j), pW(g)} and θW(j− g) ≥ min{θW(j), θW(g)} , qW(j− g) ≥ min{qW(j), qW(g)} and φW(j− g) ≥ min{φW(j), φW(g)} , rW(j− g) ≤ max{rW(j), rW(g)} , ωW(j− g) ≤ max{ωW(j), ωW(g)} . pW(jg) ≥ α and θW(jg) ≥ α̂, qW(jg) ≥ β and φW(jg) ≥ β̂ rW(jg) ≤ γ and ωW(jg) ≤ γ̂ Implies that pW(jg) ≥ min{pW(j), pW(g)} and θW(jg) ≥ min{θW(j), θW(g)} qW(jg) ≥ min{qW(j), qW(g)} and φW(jg) ≥ min{φW(j), φW(g)} rW(jg) ≤ max{rW(j), rW(g)} and ωW(jg) ≤ max{ωW(j), ωW(g)} . Thus, TW(j− g) = pW(j− g)eiθW(j−g) ≥ min{pW(j), pW(g)} eimin{θW(j),θW(g)} = min{pW(j)eiθW(j), pW(g)eiθW(g)} M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 15 of 23 TW(j− g) ≥ min{TW(j),TW(g)} . IW(j− g) = qW(j− g)eiφW(j−g) ≥ min{qW(j), qW(g)} eimin{φW(j),φW(g)} = min{qW(j)eiφW(j), qW(g)eiφW(g)} IW(j− g) ≥ min{IW(j), IW(g)} . FW(j− g) = rW(j− g)eiωW(j−g) ≤ max{rW(j), rW(g)} eimax{ωW(j),ωW(g)} = max{rW(j)eiωW(j), rW(g)eiωW(g)} FW(j− g) ≤ max{FW(j),FW(g)} TW(jg) = pW(jg)eiθW(jg) ≥ min{pW(j), pW(g)} eimin{θW(j),θW(g)} = min{pW(j)eiθW(j), pW(g)eiθW(g)} TW(jg) ≥ min{TW(j),TW(g)} . IW(jg) = qW(jg)eiφW(jg) ≥ min{qW(j), qW(g)} eimin{φW(j),φW(g)} = min{qW(j)eiφW(j), qW(g)eiφW(g)} IW(jg) ≥ min{IW(j), IW(g)} . FW(jg) = rW(jg)eirW(jg) ≤ max{rW(j), rW(g)} eimax{ωW(j),ωW(g)} = max{rW(j)eiωW(j), rW(g)eiωW(g)} FW(jg) ≤ max{FW(j),FW(g)} . Further, let f ∈ H be any element. Let pW(j) = α, θW(j) = α̂, qW(j) = β, φW(j) = β̂, rW(j) = γ, and ωW(j) = γ̂. Then, pW(j) ≥ α, θW(j) ≥ α̂, qW(j) ≥ β, φW(j) ≥ β̂, and rW(j) ≤ γ̂, ωW(j) ≤ γ̂ is true. Implies that j ∈ W (α, β, γ) (α̂, β̂, γ̂) . 4. Properties of the Direct Product of Complex Neutrosophic Subrings In this part, we describe the direct product of CNSRs. We use the abstraction of CNSs to explore the fundamental properties the direct product of CNSR. Definition 11. Assume that W and X be any two π-NS of sets K1 and K2, conse- quently. The cartesian product of π-NS W and X is expressed as (Wπ × Xπ)(f, g) = {< (f, g), TWπ×Xπ(f, g), IWπ×Xπ(f, g),FWπ×Xπ(f, g) >},∀ f ∈ K1, g ∈ K2. Remark 2. Let W and X be two π-NSRs of K1 and K2, respectively. Then Wπ × Xπ is π-CNSR of K1 ×K2. Remark 3. A π-CNSR Wπ ×Xπ of ring K1 ×K2 is a π-CNSR of K1 ×K2 if and only if W× X is CNSR of K1 ×K2 M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 16 of 23 Definition 12. Let W and X be two CNSs of set P . The cartesian product of CNSSs W and X is expressed by a function W× X = {< (ϑ, ),TW×X(ϑ, ), IW×X(ϑ, ),FW×X(ϑ, ) >}, TW×X(ϑ, ) = pW×X(ϑ, )e iθW×X(ϑ,) = min{pW(ϑ), pX()}eimin{θW(ϑ),θX()}, IW×X(ϑ, ) = qW×X(ϑ, )e iϕW×X(ϑ,) = min{qW(ϑ), qX()}eimin{φW(ϑ),φX()}, FW×X(ϑ, ) = rW×X(ϑ, )e irW×X(ϑ,) = max{rW(ϑ), rX()}eimax{ωW(ϑ),ωX()}. In this paper we shall take TW×X(ϑ, ) = pW×X(ϑ, )e iθW×X(ϑ,), IW×X(ϑ, ) = qW×X(ϑ, )e iφW×X(ϑ,) and FW×X(ϑ, ) = rW×X(ϑ, )e iωW×X(ϑ,) for the level of truth, level of neutral and level of falsehood of W×X. The upcoming theorem explain that the cartesian product of two CNSRs is CNSR. Theorem 6. Let W and X be two CNSRs of R1 and R2, consequently. Then W × X is complex neutrosophic subrings of R1 ×R2. Proof. Let ϑ, k ∈ R1 and , l ∈ R2 be an elements. Then (ϑ, ), (k, l) ∈ R1 × R2. Consider TW×X((ϑ, )− (k, l)) = TW×X(ϑ− k, − l) = pW×X(ϑ− k, − l)eiθW×X(x−k,y−l) = min{pW(ϑ− k), pX(− l)} eimin{θW(ϑ−k),θX(−l)} = min{pW(ϑ− k)eiθW(ϑ−k), pX(− l)eiθW(−l)} = min{TW(ϑ− k),TX(− l)} ≥ min{min{TW(ϑ),TW(k)} ,min{TX(),TX(l)} } = min{min{TW(ϑ),TX()} ,min{TW(k),TX(l)} } ≥ min{TW×X(ϑ, ),TW×X(k, l)} TW×X((ϑ, )− (k, l)) ≥ min{TW×X(ϑ, ),TW×X(k, l)} . TW×X((ϑ, )(k, l)) = TW×X((ϑ, )(ϑ, ))TW×X((ϑ, )(k, l)) = TW×X(ϑk, l) = pW×X(ϑk, l)e iθW×X(ϑk,l) = min{pW(ϑk), pX(l)} eimin{θW(ϑk),θX(l)} = min{pW(ϑk)eiθW(ϑk), pX(l)e iθW (l)} = min{TW(ϑk),TX(l)} ≥ min{min{TW(ϑ),TW(k)} ,min{TX(),TX(l)} } = min{min{TW(ϑ),TX()} ,min{TW(k),TX(l)} } ≥ min{TW×X(ϑ, ),TW×X(k, l)} TW×X((ϑ, )(k, l)) ≥ min{TW×X(ϑ, ),TW×X(k, l)} . Consider IW×X((ϑ, )− (k, l)) = χW×X((ϑ, )(ϑ, ))IW×X((ϑ, )− (k, l)) = IW×X(ϑ− k, − l) = qW×X(ϑ− k, − l)eiφW×X(ϑ−k,−l) M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 17 of 23 = min{qW(ϑ− k), qX(− l)} eimin{φW(ϑ−k),φX(g−l)} = min{qW(ϑ− k)eiφW(ϑ−k), qX(− l)eiφW (−l)} = min{IW(ϑ− k), IX(− l)} ≥ min{min{IW(ϑ), IW(k)} ,min{IX(), IX(l)} } = min{min{IW(ϑ), IX()} ,min{IW(k), IX(l)} } ≥ min{IW×X(ϑ, ), IW×X(k, l)} IW×X((ϑ, )− (k, l)) ≥ min{IW×X(ϑ, ), IW×X(k, l)} . IW×X ( (ϑ, )(k, l) ) = IW×X(ϑk, l) = qW×X(ϑk, l)e iφW×X(ϑk,l) = min { qW(ϑk), qX(l) } eimin{φW(ϑk), φX(l)} = min { qW(ϑk)eiφW(ϑk), qX(l)e iφX(l) } = min { IW(ϑk), IX(l) } ≥ min { min{IW(ϑ), IW(k)},min{IX(), IX(l)} } ≥ min { IW×X(ϑ, ), IW×X(k, l) } . Assume that, FW×X((ϑ, )− (k, l)) = FW×X(ϑ− k, − l) = rW×X(ϑ− k, − l)eiωW×X(ϑ−k,−l) = max{rW(ϑ− k), rX(− l)} eimax{ωW(ϑ−k),ωX(−l)} = max{rW(ϑ− k)eiωW(ϑ−k), rX(− l)eiωW(−l)} = max{FW(ϑ− k),FX(− l)} ≤ max{max{FW(ϑ),FW(k)} ,max{FX(),FX(l)} } = max{max{FW(ϑ),FX()} ,max{FW(k),FX(l)} } ≤ max{FW×X(ϑ, ),FW×X(k, l)} FW×X((ϑ, )− (k, l)) ≤ max{FW×X(ϑ, ),FW×X(k, l)} . Now, we take, FW×X((ϑ, )(k, l)) = FW×X(ϑk, l) = rW×X(ϑk, l)e iωW×X(ϑk,l) = max{rW(ϑk), rX(l)} ∗ eimax{ωW(ϑk),ωX(l)} = max{rW(ϑk)eiωW(ϑk), rX(l)e iωW(l)} = max{FW(ϑk),FX(l)} ≤ max{max{FW(ϑ),FW(k)} ,max{FX(),FX(l)} } = max{max{FW(ϑ),FX()} ,max{FW(k),FX(l)} } ≤ max{FW×X(ϑ, ),FW×X(k, l)} FW×X((ϑ, )(k, l)) ≤ max{FW×X(ϑ, ),FW×X(k, l)}. Hence the desired result is obtained. M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 18 of 23 Corollary 1. Let W1, W2, . . . ,Wn be CNSRs of K1, K2, . . . ,Kn, respectively. Then W1 ×W2×, . . . ,×Wn is CNSR of K1 ×K2 × . . . ×Kn. Remark 4. Let W and X be two CNSRs of K1 and K2, consequently and W1 ×W2 be CNSR of K1 ×K2. Then it is not compulsory both W1 and W2 should be CNSRs of K1 and K2, consequently. Remark 5. Let W × X be CNSR of ring K1 × K2. Then pW×X(0, 0 ′ ) ≥ pW×X(f, g), θW×X(0, 0 ′ ) ≥ θW×X(f, g), qW×X(0, 0 ′ ) ≥ qW×X(f, g), φW×X(0, 0 ′ ) ≥ φW×X(f, g), rW×X(0, 0 ′ ) ≤ rW×X(f, g), and ωW×X(0, 0 ′ ) ≤ ωW×X(f, g). ∀f ∈ K1, g ∈ K2. Here 0 and 0 ′ are neutral elements of K1 and K2, respectively. Theorem 7. Let W and X be two CNS of rings K1 and K2. If W × X is a complex neutrosophic subring of K1 ×K2, then one of the following statements must be satisfied. (i) pW(0) ≥ pX(g), θW(0) ≥ θX(g), qW(0) ≥ qX(g), φW(0) ≥ φX(g) and rW(0) ≤ rX(g), ωW(0) ≤ ωX(g), ∀g ∈ K2. (ii) pX(0 ′ ) ≥ pW(f), θX(0 ′ ) ≥ θW(f), qX(0 ′ ) ≥ qW(f), φX(0 ′ ) ≥ φW(f), rX(0 ′ ) ≤ rW(f), ωX(0 ′ ) ≤ ωW(f), ∀f ∈ K1. Here 0 and 0 ′ are neutral elements of K1 and K2. Proof. Let W× X be a CNSR of K1 ×K2. On contrary, assume that the statements [1] and [2] do not hold. Then there exist f ∈ K1 and g ∈ K2 such that (i) pW(0) ≤ pX(g), θW(0) ≤ θX(g), qW(0) ≤ qX(g), φW(0) ≤ φX(g) and rW(0) ≥ rX(g), ωW(0) ≥ ωX(g), ∀g ∈ K2. (ii) pX(0 ′ ) ≤ pW(f), θX(0 ′ ) ≤ θW(f), qX(0 ′ ) ≤ qW(f), φX(0 ′ ) ≤ φW(f), rX(0 ′ ) ≥ rW(f), ωX(0 ′ ) ≥ ωW(f), ∀f ∈ K1. Consider, TW×X(f, g) = min{pW(f), pX(g)} eimin{θW(f),θX(g)} ≥ min{pW(0), pX(0 ′ )} eimin{θW(0),θX(0 ′ )} = TW×X(0, 0 ′ ). IW×X(f, g) = min{qW(f), qX(g)} eimin{φW(f),φX(g)} ≥ min{qW(0), qX(0 ′ )}eimin{φW(0),φX(0 ′ )} = IW×X(0, 0 ′ ) and FW×X(f, g) = max{rW (f), rX(g)} eimax{ωW (f),ωX(g)} ≤ max{rW(0), rX(0 ′ )}eimax{ωW(0),ωX(0 ′ )} = FW×X(0, 0 ′ ). But W × X is CNSR. Hence, it is proved that at least one statements must be satisfied. (i) pW(0) ≥ pX(g), θW(0) ≥ θX(g), qW(0) ≥ qX(g), φW(0) ≥ φX(g) and rW(0) ≤ rX(g), ωW(0) ≤ ωX(g), ∀g ∈ K2. (ii) pX(0 ′ ) ≥ pW(f), θX(0 ′ ) ≥ θW(f), qX(0 ′ ) ≥ qW(f), φX(0 ′ ) ≥ φW(f), rX(0 ′ ) ≤ rW(f), ωX(0 ′ ) ≤ ωW(f), ∀f ∈ K1. Theorem 8. Let W and X be CNSs of K1, K2 and pX(0 ′ ) ≥ pW(ϑ), θX(0 ′ ) ≥ θW(ϑ), qX(0 ′ ) ≥ qW(ϑ), φX(0 ′ ) ≥ φW(ϑ), rX(0 ′ ) ≤ rW(ϑ), ωX(0 ′ ) ≤ ωW(ϑ), ∀ ϑ ∈ K1, 0 ′ is identity of K2. If W× X is CNSR of K1 ×K2, then W is CNSR of K1. M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 19 of 23 Proof. Let (ϑ, 0 ′ ), (a, 0 ′ ) be elements of K1 ×K2. By given condition pX(0 ′ ) ≥ pW(ϑ) and θX(0 ′ ) ≥ θW(ϑ), for all ϑ, a ∈ K1 and 0 ′ ∈ K2. Consider, TW(ϑ− a) = pW(ϑ− a)eiθW(ϑ−a) = min{pW(ϑ− a)eiθW(ϑ−a), pX(0 ′ − 0 ′ )eiθX(0 ′−0 ′ )} = pW×X((ϑ, 0 ′ )(ϑ, 0 ′ ))eiθW×X((ϑ,0 ′ )(ϑ,0 ′ )) ≥ min{pW×X(ϑ, 0 ′ ), pW×X(ϑ, 0 ′ )} eimin{θW×X(a,0 ′ ),θW×X(ϑ,0 ′ )} = min{min{pW(ϑ), pX(0 ′ )} ,min{pW(a), pX(0 ′ )} } ∗ eimin{min{θX(0 ′ )}, min{θ,min{θW(a),θX(0 ′ )} } = min{TW(ϑ),TW(a)}. Thus, TW(ϑ− a) ≥ min{TW(ϑ),TW(a)} . TW(ϑa) = pW(ϑa)eiθW(ϑa) = min{pW(ϑa)eiθW(ϑa), pX(0 ′ 0 ′ )eiθX(0 ′ 0 ′ )} = pW×X((ϑ, 0 ′ )(ϑ, 0 ′ ))eiθW×X((ϑ,0 ′ )(ϑ,0 ′ )) ≥ min{pW×X(ϑ, 0 ′ ), pW×X(ϑ, 0 ′ )} eimin{θW×X(a,0 ′ ),θW×X(ϑ,0 ′ )} = min{min{pW(ϑ), pX(0 ′ )} ,min{pW(a), pX(0 ′ )} } ∗ eimin{min{θX(0 ′ )} ,min{θ,min{θW(a),θX(0 ′ )} } = min{TW(ϑ),TW(a)}. Thus, TW(ϑa) ≥ min{TW(ϑ),TW(a)} . Consider, IW(ϑ− a) = qW(ϑ− a)eiθW(ϑ−a) = min{qW(ϑ− a)eiθW(ϑ−a), qX(0 ′ − 0 ′ )eiθX(0 ′−0 ′ )} (-6) = qW×X((ϑ, 0 ′ )(ϑ, 0 ′ ))eiθW×X((ϑ,0 ′ )(ϑ,0 ′ )) ≥ min{qW×X(ϑ, 0 ′ ), qW×X(ϑ, 0 ′ )} eimin{θW×X(a,0 ′ ),θW×X(ϑ,0 ′ )} = min{min{qW (ϑ), qX(0 ′ )} ,min{qW(a), pX(0 ′ )} } ∗ eimin{min{θX(0 ′ )} ,min{θ,min{θW(a),θX(0 ′ )} } = min{IW(ϑ), IW(a)}. Thus, IW(ϑ− a) ≥ min{IW(ϑ), IW(a)} . IW(ϑa) = qW(ϑa)eiθW(ϑa) = min{qW(ϑa)eiθW(ϑa), qX(0 ′ 0 ′ )eiθX(0 ′ 0 ′ )} = qW×X((ϑ, 0 ′ )(ϑ, 0 ′ ))eiθW×X((ϑ,0 ′ )(ϑ,0 ′ )) ≥ min{qW×X(ϑ, 0 ′ ), qW×X(ϑ, 0 ′ )} eimin{θW×X(a,0 ′ ),θW×X(ϑ,0 ′ )} = min{min{qW(ϑ), qX(0 ′ )} ,min{qW(a), pX(0 ′ )} } ∗ eimin{min{θX(0 ′ )} ,min{θ,min{θW(a),θX(0 ′ )} } M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 20 of 23 = min{IW(ϑ), IW(a)}. Thus, IW(ϑa) ≥ min{IW(ϑ), IW(a)} . Further, FW(ϑ− a) = rW(ϑ− a)eiωW(ϑ−a) = {max{rW(ϑ− a)eiωW(ϑ−a), rX(0 ′ − 0 ′ )eiωX(0 ′−0 ′ )}} (-17) = {rW×X((ϑ, 0 ′ )(a, 0 ′ ))}ei{ωW×X((ϑ,0 ′ )(a,0 ′ ))} ≤ max{rW×X(ϑ, 0 ′ ), rW×X(a, 0 ′ )} eimax{ωW×X(ϑ,0 ′ ),ωW×X(a,0 ′ )} = max{max{rW (ϑ), rX(0 ′ )} ,max{rW(a), rX(0 ′ )} } ∗ eimax{max{ωX(0 ′ )}, max{ω,max{ωW(a),ωX(0 ′ )} } = max{FW(ϑ),FW(a)}. Thus, FW(ϑ− a) ≤ max{FW(ϑ),FW(a)}. Further, FW(ϑa) = rW(ϑa)eiωW(ϑa) = {max{rW(ϑa)eiωW(ϑa), rX(0 ′ 0 ′ )eiωX(0 ′ 0 ′ )}} = {rW×X((ϑ, 0 ′ )(a, 0 ′ ))}ei{ωW×X((ϑ,0 ′ )(a,0 ′ ))} ≤ max{rW×X(ϑ, 0 ′ ), rW×X(a, 0 ′ )} eimax{ωW×X(ϑ,0 ′ ),ωW×X(a,0 ′ )} = max{max{rW(ϑ), rX(0 ′ )} ,max{rW(a), rX(0 ′ )} } ∗ eimax{max{ωX(0 ′ )}, max{ω,max{ωW(a),ωX(0 ′ )} } = max{FW(ϑ),FW(a)}. Thus, FW(ϑa) ≤ max{FW(ϑ),FW(a)}. Hence, we obtained the result. Theorem 9. Let W and X two CNSSs of K1 and K2 such that pW(0) ≥ pX(g), qW(0) ≥ qX(g) and rW(0) ≥ rX(g), ∀g ∈ K2 and 0 is identity of K1. If W×X is CNSR of K1×K2, then X is a CNSR of K2. Proof. The proof is on similar lines as Theorem 4.6. Corollary 2. Let W and X be two CNSSs of K1 and K2, respectively. If W×X is CNSR of K1 ×K2, then W is a CNSR of K1 or X is a CNSR of K2. 5. Conclusion In this manuscript, we have discussed the complex neutrosophic subring, intersection, and level subset of complex neutrosophic subring. We have demonstrated that every com- plex neutrosophic subring generates two neutrosophic subrings and examined important aspects of this fact. We have shown that the level subset of the complex neutrosophic M. H. Mateen et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6261 21 of 23 subring forms the subring of the ring and we have talked about some of the algebraic characteristics of the level subset. In future we intend to extend this approch to subfield, submodules and BCK/BCI algebra. Also we intend to introduce applications of these defined algebraic structures to practical and theoretical problems. 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