The Structure of Neutrosophic Radical of an ideal and Primary Ideal in Neutrosophic Rings

Neutrosophic Sets and Systems, Vol. 97, 2026 University of New Mexico Murhaf Riad Alabdullah, The Structure of Neutrosophic Radical of an ideal and Primary Ideal in Neutrosophic Rings The Structure of Neutrosophic Radical of an ideal and Primary Ideal in Neutrosophic Rings Murhaf Riad Alabdullah1* 1 Faculty of Science, Department of Mathematics, University of Aleppo, Aleppo, Syria [email protected] Abstract: In this study, we define radical and primary ideals as novel types of neutrosophic substructures within a neutrosophic ring. We investigate the properties of these substructures based on the characteristics of neutrosophic rings. Finally, several examples are provided to illustrate the validity of the results. Although neutrosophic radical and primary ideals have intrinsic theoretical value, this study aims to lay the groundwork for establishing a Noether theorem on neutrosophic primary decompositions. Keywords: Neutrosophic ring, Neutrosophic primary, Neutrosophic Radical, Ideal. 1. Introduction Neutosophy represents an advanced understanding of intuitionistic fuzzy logic. This concept has significant implications for decision-making processes [1] and medical studies [2]. The applications of neutrosophy have been further explored in [3,4,5,6,7]. As a novel branch of philosophy, neutrosophy can be adapted to algebraic structures, thereby enabling a deeper comprehension and further development of these structures. The neutrosophic concept was first introduced by Smarandache in 1980. Neutrosophic structures represent a recent addition to the classification of algebraic structures, with numerous applications and developments reported, such as neutrosophic topologies [8,9] and neutrosophic rings [10,11,12,13]. Neutrosophic Sets and Systems, Vol. 97, 2026 426 Murhaf Riad Alabdullah, The Structure of Neutrosophic Radical of an ideal and Primary Ideal in Neutrosophic Rings Neutrosophic rings exhibit several intriguing characteristics and substructures, such as neutrosophic subrings and ideals. Extensive definitions and studies have been carried out on these structures [14,15,16]. In [17], the notion of the root of an AH-ideal was introduced. Abobala investigated the concept of maximal and minimal ideals in neutrosophic rings [18]. Alabdullah studied the concept of prime and completely prime ideals of neutrosophic rings [19]. In this study, we focus on ideals with the form 𝑃+𝑄𝐼, where 𝑃⊆𝑄 are ideals within the classical ring. Building upon the preceding concept, two novel types of neutrosophic substructures, namely radical ideals and primary ideals are introduced. Several theorems are established, describing their fundamental properties, which are both intriguing and analogous to those of classical radical and primary ideals, albeit with certain distinctions. This study aims to address a significant research gap by identifying and characterizing all radical and primary ideals within neutrosophic rings. Furthermore, it will contribute to the classification of specific types of neutrosophic rings, such as Noetherian and Artinian rings. 2. Definitions and notations Since researchers interested in classical rings already possess comprehensive knowledge of them and their ideals, this part presents neutrosophic rings and the properties of their ideals. Definition 2.1 [11] Assume that 𝑅 is a ring. The collection 𝑅(𝐼)={𝑎+𝑏𝐼 ;𝑎,𝑏∈𝑅 and 𝐼2=𝐼} is called a neutrosophic ring. When 𝑅 is a field, 𝑅(𝐼) is a neutrosophic field. Properties 2.2 [11] 1. A ring 𝑅 is a unity commutative ring iff 𝑅(𝐼) is a unity commutative neutrosophic ring with neutrosophic unity 𝐼. 2. 𝐼𝑚=𝐼, ∀𝑚∈ℤ+ 3. 𝑥𝐼=𝐼𝑥,∀𝑥∈𝑅. 4. 0𝐼=0 and 𝐼+𝐼+⋯+𝐼 ⏟ 𝑚 𝑡𝑖𝑚𝑒 =𝑚𝐼 Definition 2.3 [11] If ∅≠𝐽⊆𝑅(I), 𝐽 is called a neutrosophic ideal if it is a neutrosophic subring of 𝑅(I) and 𝑥𝑗,𝑗𝑥∈𝐽 𝑓𝑜𝑟 𝑒ach ,𝑗∈𝐽 𝑎𝑛𝑑 𝑥∈𝑅(𝐼). Theorem 2.4 [18] If 𝐽+𝐾𝐼⊆𝑅(I), then 𝐽+𝐾𝐼 is a neutrosophic ideal iff 𝐽 𝑎𝑛𝑑 𝐾 are ideals in 𝑅, where 𝐽⊆𝐾. Theorem 2.5 [18] If 𝐽+𝐾𝐼 is an ideal in 𝑅(I), 𝐽+𝐾𝐼 is a neutrosophic maximal ideal iff 𝐽 is a maximal ideal of 𝑅, where 𝐾=𝑅 𝑜𝑟 𝐽+𝐾𝐼=𝑅(𝐼). Definition 2.6 [19] Assume that 𝑅(I) is a neutrosophic ring and that 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I). 1. 𝐽+𝐾𝐼 is a neutrosophic prime if it satisfies the following condition: ∀𝐽1+𝐾1𝐼,𝐽2+𝐾2𝐼∈𝑁𝔗𝑅(I); 𝐽1⊆𝐾1 𝑎𝑛𝑑 𝐽2⊆𝐾2; (𝐽1+𝐾1𝐼)(𝐽2+𝐾2𝐼)⊆𝐽+𝐾𝐼 Neutrosophic Sets and Systems, Vol. 97, 2026 427 Murhaf Riad Alabdullah, The Structure of Neutrosophic Radical of an ideal and Primary Ideal in Neutrosophic Rings ⇒ 𝐽1+𝐾1𝐼⊆𝐽+𝐾𝐼 𝑜𝑟 𝐽2+𝐾2𝐼⊆𝐽+𝐾𝐼 2. 𝐽+𝐾𝐼 is a completely prime if it satisfies the following condition: ∀𝑟1+𝑟2𝐼 𝑎𝑛𝑑 𝑟3+𝑟4𝐼∈𝑅(I); (𝑟1+𝑟2𝐼)( 𝑟3+𝑟4𝐼)∈𝐽+𝐾𝐼 ⇒ 𝑟1+𝑟2𝐼∈𝐽+𝐾𝐼 ˅ 𝑟3+𝑟4𝐼∈𝐽+𝐾𝐼 Theorem 2.7 [19] Assume that 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I). If 𝐽+𝐾𝐼∈𝑁℘𝑅(𝐼),𝑡ℎ𝑒𝑛 𝐽 𝑎𝑛𝑑 𝐾∈℘𝑅. Theorem 2.8 [19] Assume that 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I). If 𝐽+𝐾𝐼∈𝑁𝐶℘𝑅(𝐼),𝑡ℎ𝑒𝑛 𝐽 𝑎𝑛𝑑 𝐾∈𝐶℘𝑅. Theorem 2.9 [19] Assume that 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I), 𝐽+𝐾𝐼∈𝑁𝐶℘𝑅(𝐼),𝑡ℎ𝑒𝑛 𝐽+𝐾𝐼∈𝑁℘𝑅(𝐼). Theorem 2.10 [19] If 𝑅(I) is a unity and 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I), 𝐽+𝐾𝐼 is a neutrosophic prime iff it satisfies the following condition: ∀𝑟1+𝑟2𝐼 𝑎𝑛𝑑 𝑟3+𝑟4𝐼∈𝑅(𝐼);(𝑟1+𝑟2𝐼)𝑅(𝐼)(𝑟3+𝑟4𝐼)⊆𝐽+𝐾𝐼 ⇒𝑟1+𝑟2𝐼∈𝐽+𝐾𝐼 𝑜𝑟 𝑟3+𝑟4𝐼∈𝐽+𝐾𝐼 Corollary 2.11 [19] Assume that 𝑅(I) is a commutative neutrosophic ring with unity. If 𝐽+𝐾𝐼∈𝑁℘𝑅(𝐼),𝑡ℎ𝑒𝑛 𝐽+𝐾𝐼∈𝑁𝐶℘𝑅(𝐼). Definition 2.12 [17] Assume that 𝑅(𝐼) is a neutrosophic ring, and let 𝑃 = 𝑃0 + 𝑃1𝐼 = {𝑎0 +𝑎1𝐼 ; 𝑎0 ∈𝑃0 ,𝑎1∈ 𝑃1}. Then 𝑃 is called an AH-ideal if 𝑃0 𝑎𝑛𝑑 𝑃1 are ideals in 𝑅. Definition 2.13 [17] Assume that 𝑅(𝐼) is a commutative and that 𝑃= 𝑃0+ 𝑃1𝐼 is an AH-ideal. Then the AH-root of 𝑃 can be defined as: 𝐴𝐻−𝑅𝑎𝑑(𝑃)=√𝑃0+√𝑃1𝐼. We denote by 𝑁𝔗𝑅(I) the collection of all neutrosophic ideals of 𝑅(𝐼). Moreover, we use (𝑁℘𝑅(𝐼),𝑁𝐶℘𝑅(𝐼), 𝑁G℘ℜ𝑅(𝐼), 𝑁℘ℜ𝑅(𝐼)) to denote the collections of neutrosophic (prime, completely prime, generalized primary, primary) ideals, respectively. In the classical ring 𝑅, we denote the collection of all ideals by 𝔗𝑅, and the collections of (prime, completely prime, generalized primary, primary) ideals by (℘𝑅,𝐶℘𝑅,G℘ℜ𝑅,℘ℜ𝑅), respectively. Throughout this paper, 𝑅(𝐼) is assumed to be a neutrosophic ring with unity. 3. Radical of Neutrosophic Ideals Theorem 3.1 Assume that 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I). Then 𝐽+𝐾𝐼∈𝑁℘𝑅(𝐼) iff 𝐽∈℘𝑅 𝑎𝑛𝑑 𝐾=𝑅. Proof. (⇒) if 𝐽+𝐾𝐼∈𝑁℘𝑅(𝐼), then both 𝐽 𝑎𝑛𝑑 𝐾∈℘𝑅, according to Theorem 2.7. Now, we proceed to show that 𝐾=𝑅. Suppose that 𝑘∈𝐾 We have 𝐼 𝑎𝑛𝑑 1+(𝑘−1)𝐼∈𝑅(𝐼). On the other hand, we note 𝐼𝑅(𝐼) [ 1+(𝑘−1)𝐼]=𝑅𝐼[1+(𝑘−1)𝐼]=𝑅[𝐼+𝑘𝐼−𝐼]=0+𝑅𝑘𝐼⊆𝐽+𝐾𝐼 Since 𝐽+𝐾𝐼∈𝑁℘𝑅(𝐼), it follows from Theorem 2.10 that either 𝐼∈𝐽+𝐾𝐼 𝑜𝑟 1+(𝑘−1)𝐼∈𝐽+ 𝐾𝐼. Neutrosophic Sets and Systems, Vol. 97, 2026 428 Murhaf Riad Alabdullah, The Structure of Neutrosophic Radical of an ideal and Primary Ideal in Neutrosophic Rings Suppose first that 𝐼∈𝐽+𝐾𝐼. Then 𝐼∈𝐾𝐼⇒1∈𝐾. Hence, 𝐾=𝑅. Now suppose that 1+(𝑘−1)𝐼∈𝐽+𝐾𝐼. Then, 1∈𝐽⊆𝐾 , and therefore 𝐾=𝑅. In both cases, we conclude that 𝐾=𝑅 . (⇐): Suppose that 𝐽∈℘ℜ𝑅 𝑎𝑛𝑑 𝐾=𝑅. Let 𝐽1+𝐾1𝐼,𝐽2+𝐾2𝐼; 𝐽1⊆𝐾1 𝑎𝑛𝑑 𝐽2⊆𝐾2 be neutrosophic ideals of 𝑅(𝐼), such that: (𝐽1+𝐾1𝐼)( 𝐽2+𝐾2𝐼)⊆𝐽+𝑅𝐼 ⇒𝐽1𝐽2+(𝐽1𝐾2+𝐾1𝐽2+𝐾1𝐾2)𝐼⊆𝐽+𝑅𝐼 Therefore, 𝐽1𝐽2⊆𝐽 𝑎𝑛𝑑 𝐽1𝐾2+𝐾1𝐽2+𝐾1𝐾2⊆𝑅. Since 𝐽 is the prime ideal, either 𝐽1⊆𝐽 𝑜𝑟 𝐽2⊆𝐽. If 𝐽1⊆𝐽, then 𝐽1+𝐾1𝐼⊆𝐽+𝑅𝐼. Similarly, if 𝐽2⊆𝐽, then 𝐽2+𝐾2𝐼⊆𝐽+𝑅𝐼. Therefore, 𝐽+𝐾𝐼 is a neutrosophic prime. Theorem 3.2 Assume that 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I). Then 𝐽+𝐾𝐼∈𝑁𝐶℘𝑅(𝐼) iff 𝐽∈C℘𝑅 𝑎𝑛𝑑 𝐾=𝑅. Proof. In a way analogous to the proof of Theorem 3.1. Definition 3.3 Assuming that 𝑅(I) is a neutrosophic ring and 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I). We define the radical of ideal 𝐽+𝐾𝐼 as follows: 𝑅𝑎𝑑(𝐽+𝐾𝐼)=√𝐽+𝐾𝐼={𝑟1+𝑟2𝐼∈𝑅(𝐼) ; (𝑟1+𝑟2𝐼)𝑛∈𝐽+𝐾𝐼 ,𝑛∈ℤ+} Theorem 3.4 Assume 𝑅(I) is a neutrosophic ring and that 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I). Then 𝑟1+𝑟2𝐼∈√𝐽+𝐾𝐼 𝑖𝑓𝑓 ∃𝑛∈ℤ+; 𝑟1𝑛∈𝐽 𝑎𝑛𝑑 (𝑟1+𝑟2)𝑛∈𝐾; (equivalently,𝑟1∈√𝐽 𝑎𝑛𝑑 𝑟1+𝑟2∈√𝐾 ). Proof. (⇒): Suppose that 𝑟1+𝑟2𝐼∈√𝐽+𝐾𝐼. Then, by Definition 3.3, we have 𝑟1+𝑟2𝐼∈𝑅(𝐼) 𝑎𝑛𝑑 ∃𝑛∈ ℤ+; (𝑟1+𝑟2𝐼)𝑛∈𝐽+𝐾𝐼. ⇒(𝑟1+𝑟2𝐼)𝑛=𝑟1𝑛+(∑𝐶𝑖𝑛 𝑛 𝑖=1 𝑟1𝑛−𝑖𝑟2𝑖)𝐼∈𝐽+𝐾𝐼⇒ 𝑟1𝑛∈𝐽⊆𝐾 𝑎𝑛𝑑 (∑𝐶𝑖𝑛 𝑛 𝑖=1 𝑟1𝑛−𝑖𝑟2𝑖)∈𝐾 ⇒𝑟1𝑛∈𝐽⊆𝐾 𝑎𝑛𝑑 𝑟1𝑛+(∑𝐶𝑖𝑛 𝑛 𝑖=1 𝑟1𝑛−𝑖𝑟2𝑖)=(𝑟1+𝑟2)𝑛∈𝐾 ⇒𝑟1∈√𝐽 𝑎𝑛𝑑 𝑟1+𝑟2∈√𝐾 (⇐): Suppose that ∃𝑛∈ℤ+; 𝑟1𝑛∈𝐽 𝑎𝑛𝑑 (𝑟1+𝑟2)𝑛∈𝐾. Neutrosophic Sets and Systems, Vol. 97, 2026 429 Murhaf Riad Alabdullah, The Structure of Neutrosophic Radical of an ideal and Primary Ideal in Neutrosophic Rings ⇒𝑟1 𝑛∈𝐽⊆𝐾 𝑎𝑛𝑑 (𝑟1+𝑟2)𝑛=𝑟1𝑛+(∑𝐶𝑖𝑛 𝑛 𝑖=1 𝑟1𝑛−𝑖𝑟2𝑖)∈𝐾⇒−𝑟1𝑛+𝑟1𝑛+(∑𝐶𝑖𝑛 𝑛 𝑖=1 𝑟1𝑛−𝑖𝑟2𝑖) =∑𝐶𝑖𝑛 𝑛 𝑖=1 𝑟1𝑛−𝑖𝑟2𝑖∈𝐾 On the other hand, we have (𝑟1+𝑟2𝐼)𝑛=𝑟1𝑛+(∑𝐶𝑖𝑛 𝑛 𝑖=1 𝑟1𝑛−𝑖𝑟2𝑖)𝐼∈𝐽+𝐾𝐼 Therefore, by Definition 3.3, we conclude that 𝑟1+𝑟2𝐼∈√𝐽+𝐾𝐼. Corollary 3.5 If 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I), then √𝐽+𝐾𝐼=√𝐽+√𝐾𝐼. Proof. ∀𝑟1+𝑟2𝐼∈√𝐽+𝐾𝐼⇒ 𝑟1∈√𝐽 𝑎𝑛𝑑 𝑟1+𝑟2∈√𝐾 by Theorem 3.4. Since 𝐽 ⊆𝐾, it follows that √𝐽 ⊆ √𝐾. Now, since 𝑟1∈√𝐽 ⊆√𝐾 Since 𝑟1∈√𝐽 ⊆√𝐾, we also have −𝑟1∈√𝐾. Therefore, 𝑟1∈√𝐽 𝑎𝑛𝑑 𝑟2∈√𝐾. Hence, √𝐽+𝐾𝐼⊆√𝐽+ √𝐾𝐼. Conversely, ∀𝑟1+𝑟2𝐼∈√𝐽+√𝐾𝐼⇒ 𝑟1∈√𝐽⊆√𝐾 𝑎𝑛𝑑 𝑟2∈√𝐾⇒𝑟1+𝑟2∈√𝐾 By Theorem 3.4, it follows that 𝑟1+𝑟2𝐼∈√𝐽+𝐾𝐼. Therefore, √𝐽+√𝐾𝐼⊆√𝐽+𝐾𝐼. Thus, equality is achieved. Corollary 3.6 It is clear from Corollary 3.5 that radical of any ideal 𝐽+𝐾𝐼 is also an 𝐴𝐻− 𝑅𝑎𝑑(𝐽+𝐾𝐼). Theorem 3.7 Assume that 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I). Then √𝐽+𝐾𝐼∈𝑁𝔗𝑅(I); 𝐽+𝐾𝐼⊆√𝐽+𝐾𝐼. Proof. First, by Theorem 3.5, we have √𝐽+𝐾𝐼=√𝐽+√𝐾𝐼. Moreover, √𝐽 is an ideal containing 𝐽, and √𝐾 is an ideal containing 𝐾. Therefore, √𝐽+𝐾𝐼=√𝐽+√𝐾𝐼 is an ideal containing 𝐽+𝐾𝐼. Corollary 3.8 The neutrosophic nilpotent elements in 𝑅(I) are given by √〈0〉+〈0〉𝐼= {𝑟1+𝑟2𝐼∈𝑅(𝐼) ; (𝑟1+𝑟2𝐼)𝑛=0,𝑛∈ℤ+ }. Examples 3.9 (1) In ℤ6(𝐼), we have √ <3>+<3>𝐼 ={𝑟1+𝑟2𝐼∈ℤ6(𝐼); 𝑟1∈√<3> 𝑎𝑛𝑑 𝑟2∈√<3>} by Corollary 3.5. Since <3>∈℘ℤ6, it follows that √<3>=<3>={0,3}. Therefore, √ <3>+<3>𝐼 = {0,0,3,3𝐼,3+3𝐼}=<3>+<3>𝐼 . It is clear that √ <3>+<3>𝐼 ∈𝑁𝔗ℤ6(𝐼) and <3>+<3> 𝐼⊆√ <3>+<3>𝐼 . (2) In ℤ(𝐼), we have √ <0>+<2>𝐼 ={𝑟1+𝑟2𝐼∈ℤ(𝐼); 𝑟1∈√<0> 𝑎𝑛𝑑 𝑟2∈√<2>} by Corollary 3.5. Since <0> 𝑎𝑛𝑑 <2>∈℘ ℤ, it follows that √<0>={0} 𝑎𝑛𝑑 √<2>=<2>. Therefore, √ <0>+<2>𝐼 =<0>+<2>𝐼. It is clear that √ <0>+<2>𝐼 ∈𝑁𝔗ℤ(I) and <0>+<2>𝐼 ⊆√ <0>+<2>𝐼 . Neutrosophic Sets and Systems, Vol. 97, 2026 430 Murhaf Riad Alabdullah, The Structure of Neutrosophic Radical of an ideal and Primary Ideal in Neutrosophic Rings (3) In ℤ5(𝐼), we have √ <0>+<1>𝐼 ={𝑟1+𝑟2𝐼∈ℤ5(𝐼); 𝑟1√<0>𝑎𝑛𝑑 𝑟2∈√<1>} by Corollary 3.5. Since √<0>={0} 𝑎𝑛𝑑 √<1>=<1>=ℤ5, it follows that √ <0>+<1>𝐼 =<0>+<1> 𝐼=ℤ5𝐼. Corollary 3.10 If 𝐽1+𝐾1𝐼 𝑎𝑛𝑑 𝐽2+ 𝐾2𝐼 ∈𝑁𝔗𝑅(I), then: 1. 𝐽1+𝐾1𝐼⊆𝐽2+ 𝐾2𝐼 ⇒ √𝐽1+𝐾1𝐼⊆√𝐽2+ 𝐾2𝐼 2. √(𝐽1+𝐾1𝐼)(𝐽2+ 𝐾2𝐼)=√(𝐽1+𝐾1𝐼)∩(𝐽2+ 𝐾2𝐼)=√𝐽1+𝐾1𝐼∩√𝐽2+ 𝐾2𝐼 3. √√𝐽1+𝐾1𝐼=√𝐽1+𝐾1𝐼 4. √(𝐽1+𝐾1𝐼)+(𝐽2+ 𝐾2𝐼) =√√𝐽1+𝐾1𝐼+√𝐽2+ 𝐾2𝐼 Proof. (1) ∀𝑗1+𝑘1𝐼∈√𝐽1+𝐾1𝐼⇒∃𝑛∈ℤ+ ; (𝑗1+𝑘1𝐼)𝑛∈𝐽1+𝐾1𝐼⊆𝐽2+ 𝐾2𝐼⇒𝑗1+𝑘1𝐼∈√𝐽2+ 𝐾2𝐼 ⇒√𝐽1+𝐾1𝐼⊆√𝐽2+ 𝐾2𝐼 (2) We have (𝐽1+𝐾1𝐼)(𝐽2+𝐾2𝐼)⊆𝐽1+𝐾1𝐼 𝑎𝑛𝑑 (𝐽1+𝐾1𝐼)(𝐽2+𝐾2𝐼)⊆𝐽2+𝐾2𝐼 ⇒(𝐽1+𝐾1𝐼)(𝐽2+𝐾2𝐼)⊆(𝐽1+𝐾1𝐼)∩(𝐽2+𝐾2𝐼) Hence, by (1), we obtain √(𝐽1+𝐾1𝐼)(𝐽2+ 𝐾2𝐼)⊆√(𝐽1+𝐾1𝐼)∩(𝐽2+ 𝐾2𝐼) ......(𝑖) On the other hand, (𝐽1+𝐾1𝐼)∩(𝐽2+𝐾2𝐼)⊆𝐽1+𝐾1𝐼 𝑎𝑛𝑑 (𝐽1+𝐾1𝐼)∩(𝐽2+𝐾2𝐼)⊆𝐽2+𝐾2𝐼 Hence, by (1), we obtain √(𝐽1+𝐾1𝐼)∩(𝐽2+𝐾2𝐼)⊆√𝐽1+𝐾1𝐼 𝑎𝑛𝑑 √(𝐽1+𝐾1𝐼)∩(𝐽2+𝐾2𝐼)⊆ √𝐽2+ 𝐾2𝐼 ⇒√(𝐽1+𝐾1𝐼)∩(𝐽2+𝐾2𝐼)⊆√𝐽1+𝐾1𝐼∩√𝐽2+ 𝐾2𝐼…..(𝑖𝑖) From (𝑖) and (𝑖𝑖), we obtain √(𝐽1+𝐾1𝐼)(𝐽2+ 𝐾2𝐼)⊆√(𝐽1+𝐾1𝐼)∩(𝐽2+ 𝐾2𝐼)⊆√𝐽1+𝐾1𝐼∩ √𝐽2+ 𝐾2𝐼 Now, let us prove that √𝐽1+𝐾1𝐼∩√𝐽2+ 𝐾2𝐼⊆√(𝐽1+𝐾1𝐼)(𝐽2+𝐾2𝐼). ∀𝑠+𝑡𝐼∈√𝐽1+𝐾1𝐼∩√𝐽2+ 𝐾2𝐼⇒𝑠+𝑡𝐼∈√𝐽1+𝐾1𝐼 𝑎𝑛𝑑 𝑠+𝑡𝐼∈√𝐽2+ 𝐾2𝐼 ⇒∃𝑚,𝑙∈ℤ+; (𝑠+𝑡𝐼)𝑚∈𝐽1+𝐾1𝐼 𝑎𝑛𝑑 (𝑠+𝑡𝐼)𝑙∈𝐽2+ 𝐾2𝐼 ⇒(𝑠+𝑡𝐼)𝑚(𝑠+𝑡𝐼)𝑙∈(𝐽1+𝐾1𝐼)(𝐽2+𝐾2𝐼)⇒(𝑠+𝑡𝐼)𝑚+𝑙 ∈(𝐽1+𝐾1𝐼)(𝐽2+𝐾2𝐼); 𝑚+𝑙∈ℤ+ ⇒𝑠+𝑡𝐼∈√(𝐽1+𝐾1𝐼)(𝐽2+𝐾2𝐼) Thus, equality is achieved. (3) According to Corollary 3.5, we have √√𝐽1+𝐾1𝐼=√√𝐽1+√𝐾1𝐼=√√𝐽1+√√𝐾1𝐼=√𝐽1+ √𝐾1𝐼=√𝐽1+𝐾1𝐼 . (4) According to Corollary 3.5, we have √(𝐽1+𝐾1𝐼)+(𝐽2+𝐾2𝐼)=√(𝐽1+𝐽2)+(𝐾1+𝐾2)𝐼=√𝐽1+𝐽2+√𝐾1+𝐾2𝐼……(𝑖) On the other hand, √√𝐽1+𝐾1𝐼+√𝐽2+ 𝐾2𝐼=√(√𝐽1+√𝐾1𝐼)+(√𝐽2+√𝐾2𝐼)=√(√𝐽1+√𝐽2)+(√𝐾1+√𝐾2)𝐼= √(√𝐽1+√𝐽2)+√(√𝐾1+√𝐾2)𝐼=√𝐽1+𝐽2+√𝐾1+𝐾2𝐼……(𝑖𝑖) Neutrosophic Sets and Systems, Vol. 97, 2026 431 Murhaf Riad Alabdullah, The Structure of Neutrosophic Radical of an ideal and Primary Ideal in Neutrosophic Rings From (𝑖) and (𝑖𝑖), equality is achieved. Example 3.11 (1) In ℤ(𝐼), we have <16>+<4>𝐼⊆<8>+<4>𝐼. First, we note √<16>+<4>𝐼={𝑟1+𝑟2𝐼∈ℤ(𝐼); 𝑟1∈√<16>=<2> 𝑎𝑛𝑑 𝑟2∈√<4>=< 2>}=<2>+<2>𝐼. On the other hand, we also have √<8>+<4>𝐼={𝑟1+𝑟2𝐼∈ℤ(𝐼); 𝑟1∈√<8>=<2> 𝑎𝑛𝑑 𝑟2∈ √<4>=<2>}=<2>+<2>𝐼. Therefore, it follows that √<16>+<4>𝐼⊆√<8>+<4>𝐼. (2) In ℤ6(𝐼), we have <3>+<1>𝐼 𝑎𝑛𝑑<3>+<3>𝐼∈𝑁𝔗ℤ6(𝐼). First, we note √(<3>+<1>𝐼)∩(<3>+<3>𝐼)=√<3>+<3>𝐼=<3>+<3>𝐼. On the other hand, we also have √<3>+<1>𝐼∩√<3>+<3>𝐼=(<3>+<1>𝐼)∩ (<3>+<3>𝐼)=<3>+<3>𝐼. Finally, √(<3>+<1>𝐼)(<3>+<3>𝐼)=√<3>+<3>𝐼=<3>+<3>𝐼. (3) In ℤ8(𝐼), we have √√<4>+<2>𝐼=√<2>+<2>𝐼=<2>+<2>𝐼=√<4>+<2>𝐼. (4) In ℤ6(𝐼), we have √(<2>+<1>𝐼)+(<2>+<2>𝐼)=√<2>+<1>𝐼=<2>+<1>𝐼. On the other hand, we have √<2>+<1>𝐼+√<2>+<2>𝐼=(<2>+<1>𝐼)+(<2>+<2>𝐼)=<2>+<1>𝐼. Theorem 3.12 Assume that 𝑆+𝑇𝐼∈𝑁𝔗𝑅(𝐼). Then 𝑆+𝑇𝐼∈𝑁𝐶℘𝑅(𝐼) 𝑖𝑓𝑓 √𝑆+𝑇𝐼=𝑆+𝑇𝐼=𝑆+𝑅𝐼. Proof. (⇒): According to Corollary 3.5, we have √ 𝑆+𝑇𝐼=√𝑆+√𝑇𝐼. Since 𝑆+𝑇𝐼∈𝑁𝐶℘𝑅(𝐼), it follows that 𝑆∈𝐶℘𝑅, and 𝑇=𝑅, by Theorem 3.2. Since 𝑆∈𝐶℘𝑅, it follows that √𝑆=𝑆. Thus, √ 𝑆+𝑇𝐼=√𝑆+√𝑅𝐼=𝑆+𝑅𝐼. (⇐): ∀𝑟1+𝑟2𝐼∈ 𝑅(𝐼); (𝑟1+𝑟2𝐼)2∈𝑆+𝑇𝐼⇒𝑟1+𝑟2𝐼∈√𝑆+𝑇𝐼=𝑆+𝑇𝐼⇒𝑟1+𝑟2𝐼∈𝑆+𝑇𝐼 Thus, 𝑆+𝑇𝐼∈𝑁𝐶℘𝑅(𝐼). Corollary 3.13 Assume that 𝑅(𝐼) is a commutative neutrosophic ring. Then 𝑆+𝑇𝐼∈ 𝑁℘𝑅(𝐼) 𝑖𝑓𝑓 √𝑆+𝑇𝐼= 𝑆+𝑇𝐼. Proof. Since 𝑅(𝐼) is a commutative, the result follows directly from Corollary 2.11 and Theorem 3.12. Example 3.14 We have <3>+𝑍6𝐼={0,3}+𝑍6𝐼∈𝑁℘𝑍6(𝐼), and we note that √<3>+𝑍6𝐼=<3>+𝑍6𝐼. Corollary 3.15 If 𝐽+𝐾𝐼∈𝑁𝐶℘𝑅(𝐼), then √ (𝐽+𝐾𝐼)𝑛= 𝐽+𝐾𝐼 ∀𝑛∈ℤ+. Proof. We apply the principle of mathematical induction. If 𝑛=1, then by Theorem 3.12, we have √ 𝐽+𝐾𝐼= 𝐽+𝐾𝐼. Assume that for all 𝑙∈ℤ+ with 1≤𝑙<𝑛, we have √ (𝐽+𝐾𝐼)𝑙= 𝐽+𝐾𝐼. We now prove that the equality holds for 𝑛=𝑙+1, i.e., we show that √ (𝐽+𝐾𝐼)𝑙+1 = 𝐽+𝐾𝐼. Using Corollary 3.10, we obtain the following: √ (𝐽+𝐾𝐼)𝑙+1 =√(𝐽+𝐾𝐼)𝑙(𝐽+𝐾𝐼)=√(𝐽+𝐾𝐼)𝑙∩(𝐽+𝐾𝐼)=√ (𝐽+𝐾𝐼)𝑙∩√(𝐽+𝐾𝐼) =(𝐽+𝐾𝐼) ∩(𝐽+𝐾𝐼)=𝐽+𝐾𝐼 Neutrosophic Sets and Systems, Vol. 97, 2026 432 Murhaf Riad Alabdullah, The Structure of Neutrosophic Radical of an ideal and Primary Ideal in Neutrosophic Rings Theorem 3.16 If 𝐽+𝐾𝐼 is a neutrosophic ideal in 𝑅(𝐼), then √ 𝐽+𝐾𝐼⊆{⋂𝑆𝑙+𝑇𝑙𝐼 𝑙∈𝐿; 𝑆𝑙+𝑇𝑙𝐼∈𝑁𝐶℘𝑅(𝐼) 𝑎𝑛𝑑 𝐽+𝐾𝐼⊆𝑆𝑙+𝑇𝑙𝐼} Proof. Since 𝐽+𝐾𝐼⊆𝑆𝑙+𝑇𝑙𝐼 ∀𝑙∈𝐿, it follows that √ 𝐽+𝐾𝐼⊆√𝑆𝑙+𝑇𝑙𝐼 according to Corollary 3.10. On the other hand, we have 𝑆𝑙+𝑇𝑙𝐼∈𝑁𝐶℘𝑅(𝐼), and by applying Theorem 3.12, we find that √𝑆𝑙+𝑇𝑙𝐼= 𝑆𝑙+𝑇𝑙𝐼 for all 𝑙∈𝐿. Therefore, √ 𝐽+𝐾𝐼⊆⋂𝑆𝑙+𝑇𝑙𝐼 𝑙∈𝐿. Corollary 3.17 In fact, the equality in Theorem 3.15 does not necessarily hold in the general case. Example 3.18 In ℤ(𝐼), we have √<4>+<4>𝐼=√<4>+√<4>𝐼=<2>+<2>𝐼. On the other hand, we only have that <2>+ℤ𝐼∈𝑁𝐶℘ℤ(𝐼) containing √<4>+<4>𝐼. Theorem 3.19 If 𝐽+𝑅𝐼 is a neutrosophic ideal in 𝑅(𝐼), then 𝑅𝑎𝑑( 𝐽+𝑅𝐼)=√ 𝐽+𝑅𝐼={⋂𝑆𝑙+𝑅𝐼 𝑙∈𝐿; 𝑆𝑙∈𝐶℘𝑅 𝑎𝑛𝑑 𝐽+𝑅𝐼⊆𝑆𝑙+𝑅𝐼} Proof. By Corollary 3.5, we have √ 𝐽+𝑅𝐼=√ 𝐽+√ 𝑅𝐼=√ 𝐽+𝑅𝐼. In 𝑅, we have √ 𝐽={⋂𝑆𝑙𝑙∈𝐿; 𝑆𝑙∈𝐶℘𝑅 𝑎𝑛𝑑 𝐽⊆𝑆𝑙}. Therefore, in 𝑅(𝐼), we obtain √ 𝐽+𝑅𝐼= {⋂𝑆𝑙+𝑅𝐼 𝑙∈𝐿; 𝑆𝑙∈𝐶℘𝑅 𝑎𝑛𝑑 𝐽+𝑅𝐼⊆𝑆𝑙+𝑅𝐼} Corollary 3.20 If 𝐽+𝑅𝐼 is a neutrosophic ideal in 𝑅(𝐼), then √ 𝐽+𝑅𝐼 is the smallest completely prime ideal in 𝑅(𝐼) containing 𝐽+𝑅𝐼. Proof. Using Theorem 3.19, we have √ 𝐽+𝑅𝐼={⋂𝑆𝑙+𝑅𝐼 𝑙∈𝐿; 𝑆𝑙∈𝐶℘𝑅 𝑎𝑛𝑑 𝐽+𝑅𝐼⊆𝑆𝑙+𝑅𝐼}⇒√ 𝐽+𝑅𝐼∈𝑁𝐶℘𝑅(𝐼) 𝑎𝑛𝑑 𝐽+𝑅𝐼⊆ √ 𝐽+𝑅𝐼. Assume 𝐴+𝐵𝐼∈𝑁𝐶℘𝑅(𝐼) 𝑎𝑛𝑑 𝐽+𝑅𝐼⊆𝐴+𝐵𝐼⊆√ 𝐽+𝑅𝐼. Using Corollary 3.10 and Theorem 3.12, we obtain √ 𝐽+𝑅𝐼⊆√ 𝐴+𝐵𝐼⊆√ √ 𝐽+𝑅𝐼=√ 𝐽+𝑅𝐼, which implies that √ 𝐽+𝑅𝐼=√ 𝐴+𝐵𝐼=𝐴+𝐵𝐼. Thus, √ 𝐽+𝑅𝐼 is the smallest completely prime ideal in 𝑅(𝐼) containing 𝐽+𝑅𝐼. 4. Neutrosophic Primary Ideal Definition 4.1 If 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I), then 𝐽+𝐾𝐼 is a neutrosophic generalized primary ideal if it satisfies the following condition: ∀𝐽1+𝐾1𝐼,𝐽2+𝐾2𝐼∈𝑁𝔗𝑅(I); 𝐽1⊆𝐾1 𝑎𝑛𝑑 𝐽2⊆𝐾2; (𝐽1+𝐾1𝐼)(𝐽2+𝐾2𝐼)⊆𝐽+𝐾𝐼 ⇒ 𝐽1+𝐾1𝐼⊆𝐽+𝐾𝐼 ⋁ ∃𝑛∈ℤ+; (𝐽2+𝐾2𝐼)𝑛⊆𝐽+𝐾𝐼 Neutrosophic Sets and Systems, Vol. 97, 2026 433 Murhaf Riad Alabdullah, The Structure of Neutrosophic Radical of an ideal and Primary Ideal in Neutrosophic Rings Definition 4.2 If 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I), then 𝐽+𝐾𝐼 is a neutrosophic primary ideal if it satisfies the following condition: ∀𝑟1+𝑟2𝐼 𝑎𝑛𝑑 𝑟3+𝑟4𝐼 ∈𝑅(I); (𝑟1+𝑟2𝐼)( 𝑟3+𝑟4𝐼)∈𝐽+𝐾𝐼 ⇒ 𝑟1+𝑟2𝐼∈𝐽+𝐾𝐼 ˅ ∃𝑛∈ℤ+; (𝑟3+𝑟4𝐼)𝑛∈𝐽+𝐾𝐼 Theorem 4.3 If 𝐽+𝐾𝐼∈𝑁℘ℜ𝑅(𝐼),then 𝐽+𝐾𝐼∈𝑁𝐺℘ℜ𝑅(𝐼). Proof. Assume that 𝐽+𝐾𝐼∈𝑁℘ℜ𝑅(𝐼), and let 𝐽1+𝐾1𝐼, 𝐽2+𝐾2𝐼∈𝑁𝔗𝑅(I) such that, (𝐽1+𝐾1𝐼)(𝐽2+ 𝐾2𝐼)⊆𝐽+𝐾𝐼. Now, if 𝐽1+𝐾1𝐼⊈𝐽+𝐾𝐼 𝑎𝑛𝑑 ∀𝑛∈ℤ+; (𝐽2+𝐾2𝐼)𝑛⊈𝐽+𝐾𝐼 , then ∃𝑗1+𝑘1𝐼∈ 𝐽1+𝐾1𝐼 𝑎𝑛𝑑 ∃𝑛∈ℤ+; (𝑗2+𝑘2𝐼)𝑛∈(𝐽2+𝐾2𝐼)𝑛 𝑤ℎ𝑒𝑟𝑒 𝑗1+𝑘1𝐼∉𝐽+𝐾𝐼 𝑎𝑛𝑑 (𝑗2+𝑘2𝐼)𝑛∉𝐽+ 𝐾𝐼 . On the other hand, we have (𝑗1+𝑘1𝐼)(𝑗2+𝑘2𝐼)𝑛∈(𝐽1+𝐾1𝐼)(𝐽2+𝐾2𝐼)𝑛⊆(𝐽1+𝐾1𝐼)(𝐽2+𝐾2𝐼)⊆𝐽+𝐾𝐼 Since 𝐽+𝐾𝐼∈𝑁℘ℜ𝑅(𝐼), it follows that 𝑗1+𝑘1𝐼∈𝐽+𝐾𝐼 𝑜𝑟 ∃𝑛∈ℤ+; (𝑗2+𝑘2𝐼)𝑛 ∈𝐽+𝐾𝐼. This leads to a contradiction. Therefore, 𝐽1+𝐾1𝐼⊆𝐽+𝐾𝐼 ⋁ ∃𝑛∈ℤ+; (𝐽2+𝐾2𝐼)𝑛⊆𝐽+𝐾𝐼. Thus, 𝐽+𝐾𝐼∈𝑁G℘ℜ𝑅(𝐼). Theorem 4.4 If 𝐽+𝐾𝐼∈𝑁𝔗𝑅(I), then 𝐽+𝐾𝐼 is a neutrosophic generalized primary ideal iff the following condition is satisfied: ∀ 𝑟1+𝑟2𝐼 𝑎𝑛𝑑 𝑟3+𝑟4𝐼∈𝑅(𝐼);(𝑟1+𝑟2𝐼)𝑅(𝐼)(𝑟3+𝑟4𝐼)⊆𝐽+𝐾𝐼 ⇒𝑟1+𝑟2𝐼∈𝐽+𝐾𝐼 𝑜𝑟 ∃𝑛∈ℤ+; ( 𝑟3+𝑟4𝐼)𝑛∈𝐽+𝐾𝐼 Proof. (⇒): Assume that 𝐽+𝐾𝐼 is a neutrosophic generalized primary, we prove that it satisfies the corresponding condition. ∀ 𝑟1+𝑟2𝐼 𝑎𝑛𝑑 𝑟3+𝑟4𝐼∈𝑅(𝐼); (𝑟1+𝑟2𝐼)𝑅(𝐼)(𝑟3+𝑟4𝐼)⊆𝐽+𝐾𝐼 ⇒(𝑟1+𝑟2𝐼)𝑅(𝐼)(𝑟3+𝑟4𝐼)𝑅(𝐼)⊆(𝐽+𝐾𝐼)𝑅(𝐼)⊆𝐽+𝐾𝐼 Since 𝐽+𝐾𝐼 is a neutrosophic generalized primary, then either (𝑟1+𝑟2𝐼)𝑅(𝐼)⊆𝐽+𝐾𝐼 𝑜𝑟 ∃𝑛∈ ℤ+; [(𝑟3+𝑟4𝐼)𝑅(𝐼)]𝑛⊆𝐽+𝐾𝐼. On the other hand, we have 𝑟1+𝑟2𝐼=(𝑟1+𝑟2𝐼).1∈(𝑟1+𝑟2𝐼)𝑅(𝐼)⊆𝐽+𝐾𝐼⇒𝑟1+𝑟2𝐼∈𝐽+𝐾𝐼 Also, we have [(𝑟3+𝑟4𝐼).1]𝑛∈[(𝑟3+𝑟4𝐼)𝑅(𝐼)]𝑛⊆𝐽+𝐾𝐼⇒( 𝑟3+𝑟4𝐼)𝑛∈𝐽+𝐾𝐼 (⇐): Assuming that the given condition is holds, we will now prove that 𝐽+𝐾𝐼 is a neutrosophic generalized primary. Let 𝐽1+𝐾1𝐼 𝑎𝑛𝑑 𝐽2+𝐾2𝐼 be neutrosophic ideals of 𝑅(𝐼) such that (𝐽1+𝐾1𝐼)(𝐽2+𝐾2𝐼)⊆𝐽+𝐾𝐼.