Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 345 https://internationalpubls.com Some Special Structures of 𝜹𝟏 Near-Rings 1S.Sivanthi, 2G.Sugantha, 3 M.Amirthakodi 1Research Scholar of Mathematics (Part time), Reg No: 21122102092007, Email: sssivanthi@gmail.com PG and Research Department of Mathematics, Kamaraj College, Thoothukudi – 628003 (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli – 627012) 2Assistant Professor of Mathematics, Pope’s College (Autonomous), Sawyerpuram, Tamil Nadu - 627 251, India. E.mail: sugi.trini@gmail.com (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli – 627012) 3Assistant Professor of Mathematics, PG and Research Department of Mathematics, Kamaraj College, Thoothukudi – 628003. E.mail:amirthakodim@yahoo.com (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli – 627012) Article History: Received: 14-11-2024 Revised: 25-12-2024 Accepted:09-01-2025 Abstract: If, in a ring (N, +, βˆ™) we ignore the commutativity of β€˜+’ and one of the distributive laws, (N, +, βˆ™) becomes a Near-Ring. If we do not stipulate the left distributive law, (N, +, βˆ™) is a right near- ring. This research aims to introduce the concept of Ξ΄_1 Near-Ring. For every x,y in N, xNy=Nx^2 y^2 is called Ξ΄_1Near-Ring. The element wise characterization forγ€– Ξ΄γ€—_1 Near- Ring will be investigated and shall establish theorems and properties in this Near-Ring. Mathematics Subject Classification: 16Y30 Keywords: Ξ΄_1Near-Ring, near-field. 1 Introduction A right near-ring (N, +, βˆ™) is an algebraic system with two binary operations such that (i) (N, +) is a group-not necessarily abelian-with 0 as its identity element, (ii) (N, βˆ™) is a semigroup [we write xy for x.y for all x, y in N] and (iii) (π‘₯ + 𝑦)𝑧 = π‘₯𝑧 + 𝑦𝑧 for all x, y, z in N. Because of (iii) 0n =0 for all n in N. As we do not stipulate the left distributive law, "n0 = 0” need not hold good for all n in N. We say that N is zero-symmetric if 𝑛0 = 0 for all n in N. N is called an S-near-ring or an S'-near-ring according as π‘₯ ∈ 𝑁π‘₯ or π‘₯ ∈ π‘₯𝑁 for all π‘₯ ∈ 𝑁. A subgroup M of N is called an Nsubgroup if 𝑁𝑀 βŠ‚ 𝑀 and an invariant N-subgroup if, in addition, 𝑀𝑁 βŠ‚ 𝑀. An ideal I of N is called a semi prime ideal if for all ideals J of N. 𝐽2 βŠ‚ 𝐼 β‡’ 𝐽 βŠ‚ 𝐼 . If {0} is a semiprime ideal, then N is called a semi prime near-ring. An ideal I of N is called completely semi prime if x ∈ I wheneverπ‘₯2 ∈ 𝐼. N is called a strictly prime near-ring if {0} is a strictly prime ideal i.e. if A and B are N-subgroups of N such that 𝐴𝐡 = {0}, then either 𝐴 = {0} or B = {0}. A near-ring N has property P4 if for all ideals I of N, π‘₯𝑦 ∈ 𝐼 β‡’ 𝑦π‘₯ ∈ 𝐼. From p.289 of Pilz [3] The concept of a mate function in N has been introduced in [4] with a view to handle the regularity structure in a near-ring with considerable ease. A map 𝑓 from N into N is called a mate function for N, if π‘₯ = π‘₯𝑓(π‘₯)π‘₯ for all x in N. 𝑓(π‘₯) is called a mate of x. A map 𝑓 from N into N is called a P3 mate function for N, if π‘₯ = π‘₯𝑓(π‘₯)π‘₯ and π‘₯𝑓(π‘₯) = 𝑓(π‘₯)π‘₯ for all x in N. mailto:sssivanthi@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 346 https://internationalpubls.com Basic concepts and terms used but not defined in this paper can be found in Pilz [3]. Throughout this paper N stands for a near-ring – more precisely a right near-ring – with at least two elements. As in p.249 Pilz [3], β€œif N is a near-field then either N is isomorphic to 𝑀𝑐 (𝑍2) or N is zero- symmetric” (For the concept of 𝑀𝑐 (𝑍2) one may refer to Example 1.4(a), p.8 and 1.15, p.12 of Pilz [3]. Obviously 𝑀𝑐 (𝑍2) is a near-field of order 2 and is not zero-symmetric). All the near-fields in this paper are zero-symmetric. 2 Notations (i) E denotes the set of all idempotent of N. [e in N is called an idempotent if 𝑒2 = 𝑒] (ii) L denotes the set of all nilpotent of N. [a in N is nilpotent if ak = 0 for some positive integer k.] (iii) 𝑁0 = {𝑛 ∈ 𝑁 / 𝑛0 = 0} - zero-symmetric part of N. (iv). 𝑁𝑑 = {𝑛 ∈ 𝑁 / 𝑛(π‘₯ + 𝑦) = 𝑛π‘₯ + 𝑛𝑦 for all π‘₯, 𝑦 in 𝑁} – set of all distributive element of N. (v) 𝐢(𝑁) = {𝑛 ∈ 𝑁 / 𝑛π‘₯ = π‘₯𝑛 for all π‘₯ in 𝑁} - center of N. 3. Preliminary results We freely make use of the following results from [4], [3] and [2] and designate them as K (1), K (2) etc. (K for β€˜known results’). K (1): If N has a mate function m, then for every π‘₯ ∈ 𝑁, π‘₯𝑓(π‘₯), 𝑓(π‘₯)π‘₯ ∈ E and 𝑁π‘₯ = 𝑁𝑓(π‘₯)π‘₯ and π‘₯𝑁 = π‘₯𝑓(π‘₯)𝑁. (Lemma 3.2 of [4]). K(2): If L ={0}and , N= N0 then (i) π‘₯𝑦 = 0 β‡’ 𝑦π‘₯ = 0 (for π‘₯, 𝑦 in N) and (ii) N has "Insertion of Factors Property" – IFP for short – i.e. for π‘₯, 𝑦 in N, π‘₯𝑦 = 0 β‡’ π‘₯𝑛𝑦 = 0 for all n in N. (In this paper we write that N has (*, IFP) if N has both (i) and (ii)) (Lemma 2.3 of [4]). K (3): A zero-symmetric near-ring N has IFP if and only if (0: S) is an ideal, where S is any non-empty subset of N. (9.3, p.289 of [3]). K (4): A near-ring N has no non-zero nilpotent elements if and only if π‘₯2 = 0 β‡’ π‘₯ = 0 for all x in N (Prob. 14, p.9 of [2]). 3.1 Definition and Examples In the section, we introduce the notion of 𝛿1Near-Ring and furnish examples to illustrate it. To start with we have the following definition. Definition 3.1.1 Let N be a right near-ring. If for every π‘₯, 𝑦 in 𝑁, π‘₯𝑁𝑦 = 𝑁π‘₯2𝑦2 then we say N is a 𝛿1Near-Ring. Examples 3.1.2 (i) Let (𝑁, +) be the Klein’s four group {0, π‘Ž, 𝑏, 𝑐}. The near-ring (𝑁, +,Β·) where β€˜Β·β€™ is defined as per scheme 12, p.408 of Pilz [33]. βˆ™ 0 a b c Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 347 https://internationalpubls.com 0 0 0 0 0 a 0 a 0 a b 0 0 0 0 c 0 a 0 a is a 𝛿1Near-Ring. (ii) Let (𝑁, +) be the Klein’s four group {0, π‘Ž, 𝑏, 𝑐}. The near-ring (𝑁, +,Β·) where β€˜Β·β€™ is defined as per scheme 11, p.408 of Pilz [33]. βˆ™ 0 a b c 0 0 0 0 0 a 0 a b a b 0 0 0 0 c 0 a b a is a not 𝛿1Near-Ring. Since π‘Žπ‘π‘ β‰  π‘π‘Ž2𝑏2 3.2 Properties of 𝜹𝟏 near-ring We shall obtain a complete characterization for 𝛿1near-Ring and obtain structure theorem for such near-ring – under certain conditions. Proposition 3.2.1 Let N be a 𝛿1near-ring with identity. (i) N is zero symmetric (ii) If N has no non-zero nilpotent elements then N is an S-near-ring. (iii) If N is an S'-near-ring then N has no non-zero nilpotent elements. Proof. (i) Let N be a 𝛿1near ring.Then for all x,y in N, π‘₯𝑁𝑦 = 𝑁π‘₯2𝑦2 ………….….(1) Putting y=1, we get π‘₯𝑁. 1 = 𝑁π‘₯2. 1 for all x in N. β‡’ π‘₯𝑁 = 𝑁π‘₯2 for all x in N. When π‘₯ = 0, 0𝑁 = 𝑁0 = {0}. It follows that N is zero symmetric. (ii) Putting y=1 in equation (1), we get π‘₯𝑁 = 𝑁π‘₯2 for all x in N ……………………... (2). Now π‘₯2 ∈ π‘₯𝑁 for all x in N β‡’ π‘₯2 ∈ 𝑁π‘₯2 [By equation (2)]. We have π‘₯2 = 𝑧π‘₯2 for some z in N. Therefore (π‘₯ βˆ’ 𝑧π‘₯)π‘₯ = 0. By K (2), this implies that π‘₯(π‘₯ βˆ’ 𝑧π‘₯) = 0 and 𝑧π‘₯(π‘₯ βˆ’ 𝑧π‘₯) = 0. Consequently (π‘₯ βˆ’ 𝑧π‘₯ )2 = 0 . By assumption 𝐿 = {0} therefore π‘₯ – 𝑧π‘₯ = 0 forcing π‘₯ = 𝑧π‘₯. Thus π‘₯ ∈ 𝑁π‘₯ i.e. N is an S-near-ring. (iii) If N is an S'-near-ring then π‘₯ ∈ π‘₯𝑁 and since π‘₯𝑁 = 𝑁π‘₯2 we get π‘₯ = 𝑛π‘₯2 for some 𝑛 ∈ 𝑁. Thereforeπ‘₯2 = 0 β‡’ π‘₯ = 0. N has no non-zero nilpotent elements, from K (4). Corollary3.2.2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 348 https://internationalpubls.com If N is a 𝛿1near-ring without non-zero nilpotent elements, then from K (2), we see that N has (*, IFP). It is obvious that the property 𝛿1is preserved by near-ring homomorphisms. Consequently, we have Proposition 3.2.3 Any homomorphism image of a 𝛿1near-ring is 𝛿1-near-ring. Proof Let 𝑓 : N β†’ N β€² be a near-ring epimorphism. Let π‘₯ β€², 𝑦′ ∈ 𝑁 β€². Then there exist π‘₯, 𝑦 ∈ 𝑁 such that 𝑓 (π‘₯) = π‘₯β€², 𝑓 (𝑦) = 𝑦 β€². Also for 𝑛′ ∈ 𝑁 β€² there exist 𝑛 ∈ 𝑁 such that 𝑓 (𝑛) = 𝑛′. Since N is 𝛿1 βˆ— near-ring, π‘₯𝑁𝑦 = 𝑁π‘₯2𝑦2 … … … … … (3) Now, π‘₯ ′𝑛′𝑦 β€² = 𝑓 (π‘₯)𝑓 (𝑛)𝑓 (𝑦) = 𝑓 (π‘₯𝑛𝑦) [since 𝑓 is a homomorphism] = 𝑓 (𝑛π‘₯2𝑦2) [by Equation (3)] =𝑓(𝑛)𝑓(π‘₯2)𝑓(𝑦2) Therefore, π‘₯ ′𝑛′𝑦 β€² ∈ 𝑁′π‘₯2′𝑦2β€². Consequently, π‘₯ ′𝑁 ′𝑦′ βŠ† 𝑁 β€²π‘₯2′𝑦2β€² … … … … … . . (4) Similarly, 𝑁′π‘₯2′𝑦2β€² βŠ† π‘₯ ′𝑁′𝑦 ′…………… … (5) Combining Equations (4) and (5), we get π‘₯ ′𝑁′𝑦 β€² βŠ† 𝑁′π‘₯2′𝑦2β€² Hence N β€² is also 𝛿1 near-ring and the desired result follows. As an immediate consequence of Proposition 3.2.3we have the following theorem: Theorem 3.2.4 Every 𝛿1near-ring N is isomorphic to a subdirect product of subdirectly irreducible 𝛿1 near-Ring. Proof. By Theorem 1.62, p.26 of Pilz [3], N is isomorphic to a sub direct product of sub directly irreducible near-ring Ni's, say, and each Ni is a homomorphic image of N under the projection map πœ‹π‘–. The desired result now follows from Proposition 3.2.3. We shall now discuss the behavior of N-subgroups and ideals of 𝛿1 nearRing. To start with we have the following: Proposition 3.2.5. Let N be a 𝛿1near-ring with identity, If N is left bipotent, and then every N-subgroup of N is invariant. Proof. Let N be a 𝛿1 near ring with identity. Then for all x,y in N, π‘₯𝑁 = 𝑁π‘₯2 …………….(6). Let A be any N-subgroup of N, Then 𝐴 = οƒ₯ οƒŽAx Nx …………… (7). Now, 𝑁π‘₯𝑁 = 𝑁𝑁π‘₯21 = 𝑁π‘₯2 [By equation (6)] βŠ† 𝑁π‘₯. [Since N is left bipotent] (i.e.) 𝑁π‘₯𝑁 βŠ† 𝑁π‘₯…………. (8). Therefore, 𝐴𝑁 =(οƒ₯ οƒŽAx Nx ) 𝑁[By equation (7)] βŠ† NNx Ax οƒ₯ οƒŽ βŠ†οƒ₯ οƒŽAx Nx [By equation (8)]= 𝐴 (ie.)𝐴𝑁 βŠ† 𝐴. Consequently, A is invariant N- subgroup. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 349 https://internationalpubls.com Proposition 3.2.6. Let N be a 𝛿1near-ring. Then every left ideal of N is an ideal. Proof. Let A be a left ideal of N. Since N is zero-symmetric, 𝑁𝐴 βŠ† 𝐴 i.e. A is an N-subgroup of N. Proceeding as in Proposition 3.2.5 we get 𝐴𝑁 βŠ† 𝐴. Hence A becomes an ideal. It is easy to observe the following: Corollary 3.2.7 Every left ideal (and therefore every ideal) of a 𝛿1 near-ring N is an invariant N-subgroup of N. Proposition 3.2.8. If N is a 𝛿1near-ring with identity, then N has strong IFP. Proof. Let N be a 𝛿1near ring with identity. Then for all x,y in N, π‘₯𝑁 = 𝑁π‘₯2 ……………….(9) In view of Proposition 9.2 Pilz [3], we need only to establish that for all ideals I of N and for all a, b, n in N, π‘Žπ‘ ∈ 𝐼 β‡’ π‘Žπ‘›π‘ ∈ 𝐼. Since I is an ideal, 𝐼𝑁 βŠ† 𝐼 and since N is zero-symmetric, I is an N- subgroup of N. i.e. 𝑁𝐼 βŠ† 𝐼. Now π‘Žπ‘› ∈ π‘Žπ‘ = π‘π‘Ž2 [By equation (9)] β‡’ π‘Žπ‘› = π‘›β€²π‘Ž2 for some 𝑛′ ∈ 𝑁 β‡’ π‘Žπ‘›π‘ = (π‘›β€²π‘Ž2)𝑏 = (𝑛′ π‘Ž)(π‘Žπ‘) ∈ 𝑁𝐼 β‡’ π‘Žπ‘›π‘ ∈ 𝐼. Notation 3.2.9 If a 𝛿1near-ring N is an S (or S')-near-ring then we write that N is an S - 𝛿1near-ring (or S'- 𝛿1near- ring). Remark 3.2.10 For an S- 𝛿1 near-ring, we see that for all x in N, π‘₯ ∈ 𝑁π‘₯ = π‘₯2 𝑁 β‡’ π‘₯ = π‘₯2𝑛 for some 𝑛 ∈ 𝑁. Hence π‘₯2 = 0 β‡’ π‘₯ = 0 and K (4) demands that 𝐿 = {0}. Proposition 3.2.11. In a 𝛿1 near-ring, 𝐸 βŠ† 𝐢(𝑁). Proof. Let e ∈ E. Since N is 𝛿1near-ring ,𝑒𝑁𝑒 = 𝑁𝑒2𝑒 2 = 𝑁𝑒. Therefore forsome n in N, 𝑒𝑛𝑒 = 𝑒𝑒 and 𝑛𝑒 = 𝑒𝑣𝑒 for some u, v in N. Now,𝑒𝑛𝑒 = 𝑒(𝑒𝑒) and 𝑒(𝑛𝑒) = 𝑒𝑣𝑒. Thus 𝑒𝑛𝑒 = 𝑛𝑒 for all n in N. ………….... (10).Also we have (𝑒𝑛𝑒 – 𝑒𝑛)𝑒 = 0. This implies 𝑒(𝑒𝑛𝑒 – 𝑒𝑛) = 0 and 𝑒𝑛(𝑒𝑛𝑒 – 𝑒𝑛) = 0 . Also 𝑒𝑛𝑒(𝑒𝑛𝑒 – 𝑒𝑛) = 𝑒𝑛. 0 = 0 [since N is zero-symmetric]. Now, 𝑒𝑛𝑒(𝑒𝑛𝑒 – 𝑒𝑛) βˆ’ 𝑒𝑛(𝑒𝑛𝑒 – 𝑒𝑛) = 0. Consequently, (𝑒𝑛𝑒 βˆ’ 𝑒𝑛)2 = 0 and K (4) guarantees 𝑒𝑛𝑒 βˆ’ 𝑒𝑛 = 0.Therefore,𝑒𝑛𝑒 = 𝑒𝑛 for all n in N…………….. (11).From Equations (10) and (11) we get 𝑒𝑛 = 𝑛𝑒 for all n in N. Thus 𝐸 βŠ† 𝐢(𝑁). Remark 3.2.12. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 350 https://internationalpubls.com It is worth noting that we do not stipulate that N admits mate functions for the validity of the above results. Proposition 3.2.13. Let N be a 𝛿1near-ring with identity. Then N has a mate function if and only if N is an S'-near-ring. Proof. When N has a mate function `m' for all π‘₯ ∈ 𝑁, π‘₯ = π‘₯𝑓(π‘₯)π‘₯ ∈ π‘₯𝑁 and obviously N is an S'-near-ring. Conversely let N be an S' near-ring. π‘₯ ∈ π‘₯𝑁 = 𝑁π‘₯2 β‡’ π‘₯ = 𝑛π‘₯2 for some 𝑛 ∈ 𝑁 β‡’ π‘₯2 = π‘₯𝑛π‘₯2 β‡’ (π‘₯ βˆ’ π‘₯𝑛π‘₯) π‘₯ = 0 .Using Proposition 3.2.1(iii) and Corollary 3.2.2 we get π‘₯(π‘₯ βˆ’ π‘₯𝑛π‘₯) = 0 and π‘₯𝑛π‘₯(π‘₯ βˆ’ π‘₯𝑛π‘₯) = 0 and consequently (π‘₯ βˆ’ π‘₯𝑛π‘₯)2 = 0 . Since 𝐿 = {0} we get π‘₯ βˆ’ π‘₯𝑛π‘₯ = 0i.e. π‘₯ = π‘₯𝑓(π‘₯)π‘₯ where we get 𝑓(π‘₯) = 𝑛. This guarantees that 𝑓 ∢ 𝑁 β†’ 𝑁 is a mate function for N. Proposition 3.2.14. Let N be an 𝑆’-𝛿1near-ring. Then N has a P3 mate function. Proof. When N is an S'- 𝛿1near-ring it admits a mate function β€˜π‘“β€™. From Proposition 3.2.12 we have π‘₯ = 𝑓(π‘₯) π‘₯2 β‡’ (π‘₯𝑓(π‘₯) βˆ’ 𝑓(π‘₯)π‘₯) π‘₯ = 0 β‡’ (π‘₯𝑓(π‘₯) βˆ’ 𝑓(π‘₯)π‘₯)2 = 0. [since N has (*, IFP)] π‘₯𝑓(π‘₯) βˆ’ 𝑓(π‘₯)π‘₯ = 0 β‡’ π‘₯𝑓(π‘₯) = 𝑓(π‘₯)π‘₯ i.e. 𝑓(π‘₯) ∈ 𝐢(π‘₯) i.e. β€˜π‘“β€™ is a P3 mate function. Proposition 3.2.15. If N has property 𝛿1and a mate function β€˜f’ then 𝐿 = {0} and N has (*, IFP). We now give a complete characterization of 𝛿1 when they admit mate functions. Theorem 3.2.16. Let N be a near-ring with a zero symmetric mate function β€˜π‘“β€™ and left bipotent. Then the following statements are equivalent: (i) N is 𝛿1 (ii) 𝐸 βŠ† 𝐢(𝑁) Proof. (ii) β‡’ (i) Let 𝐸 βŠ† 𝐢(𝑁) . Now 𝑁π‘₯2𝑦2 = 𝑁π‘₯𝑦2 [N is left bipotent] = (𝑁𝑓(π‘₯)π‘₯)𝑦2 [By K (1)]= (𝑓(π‘₯)π‘₯𝑁)𝑦2[Since E βŠ† C (N)] = π‘₯𝑓(π‘₯)𝑁𝑦 [N is left bipotent]= π‘₯𝑁𝑦 [By K (1)].i.e. 𝑁π‘₯2𝑦2 = π‘₯𝑁𝑦 Proof of β€˜(i) β‡’(ii)’ is similar. Remark 3.2.17 Let N admit a mate function β€˜π‘“(π‘₯)’ and let E βŠ† C (N). It is easy to observe that for every x in N,π‘₯ = π‘₯ 𝑓(π‘₯)π‘₯ β‡’ π‘₯ = 𝑓(π‘₯)π‘₯2. Consequently Proposition 3.2.16 guarantees that m is a P3 mate function. Theorem 3.2.18 Every N-subgroup of N is an ideal in an S'- 𝛿1near-ring. Proof: Since N is an S'-𝛿1near-ring, it admits a mate function β€˜f(x)’ [from Proposition 3.2.13] and L {0} [from Proposition 3.2.3(iii)]. It is clear from K (2) that N has (*, IFP). Again for any non-empty 𝑆 βŠ† 𝑁,(0 : S) is an ideal of N [by K(3)]. If M is any N-subgroup of N, then = βˆ‘ 𝑁π‘₯π‘₯βˆˆπ‘€ . We first show that each 𝑁π‘₯ is an ideal. Let 𝑆 = (0 ∢ 𝑁π‘₯). We claim that 𝑁π‘₯ = (0 ∢ 𝑆). Clearly𝑁π‘₯ βŠ‚ (0 ∢ 𝑆 )………….. (12) Now if 𝑦 ∈ (0 ∢ 𝑆) then 𝑦𝑆 = {0}. Also ( 𝑦 βˆ’ 𝑦𝑓(π‘₯)π‘₯)𝑓(π‘₯)π‘₯ = 0 ……….. (13) β‡’ ( 𝑦 βˆ’ 𝑦𝑓(π‘₯)π‘₯)𝑁𝑓(π‘₯)π‘₯ = {0} β‡’ ( 𝑦 βˆ’ 𝑦𝑓(π‘₯)π‘₯)𝑁π‘₯ = {0} [using K(1)] β‡’ ( 𝑦 βˆ’ 𝑦𝑓(π‘₯)π‘₯) ∈ (0 ∢ 𝑁π‘₯) = 𝑆. Since 𝑦𝑆 = {0}, 𝑦( 𝑦 βˆ’ 𝑦𝑓(π‘₯)π‘₯) = 0……….... (14) .Using the fact that N has (*, IFP), it is easy to Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 351 https://internationalpubls.com get from equations (13) and (14),( 𝑦 βˆ’ 𝑦𝑓(π‘₯)π‘₯)2 = 0. Since 𝐿 = {0} we get ( 𝑦 βˆ’ 𝑦𝑓(π‘₯)π‘₯) = 0 β‡’ 𝑦 = 𝑦𝑓(π‘₯)π‘₯ β‡’ 𝑦 ∈ 𝑁π‘₯ . Therefore (0 ∢ 𝑆) βŠ‚ 𝑁π‘₯…………. (15) From equations (12) and (15) we get 𝑁π‘₯ = (0 ∢ 𝑆) and hence Nx is an ideal. The desired result now follows. Remarks 3.2.19. (a) It is worth noting that in a 𝛿1near-ring with mate functions the concepts of N-subgroups, left ideals, right ideals and ideals are equivalent. (b) Recall that the nil radical of N is the greatest nil ideal of N. Since L = {0}, for an S'-𝛿1near-ring N, it follows that the nil radical of 𝑁 = {0}. Proposition 3.2.20 Let N be an S'-𝛿1near-ring. Then any N-subgroup of N is a completely semi prime ideal. Proof. Suppose I is an N-subgroup of N. From Theorem 3.2.18 it follows that I is an ideal. Let π‘₯2 ∈ 𝐼. Since N has strong IFP, π‘₯𝑓(π‘₯)π‘₯ ∈ 𝐼 i.e. π‘₯ ∈I. Hence I is a completely semi prime ideal. Proposition 3.2.21 An S'-𝛿1near-ring has property P4 Proof. Let I be an ideal of N and let π‘₯𝑦 ∈ 𝐼. Now (𝑦π‘₯)2 = (𝑦π‘₯)(𝑦π‘₯) = 𝑦(π‘₯𝑦)π‘₯ ∈ 𝑁𝐼𝑁 βŠ‚ 𝐼 [using Remark 3.2.19 (a)] β‡’ (𝑦π‘₯)2 ∈ 𝐼 .Using Proposition 3.2.20 we get 𝑦π‘₯ ∈ 𝐼. i.e.). N has property P4. 3.3 In this section we obtain a structure theorem for 𝜹𝟏 near-Ring. Throughout this section N denotes an S'-𝛿1near-ring and m is a mate function for N. Theorem 3.3.1. N is sub directly irreducible if and only if N is a near-field. Proof. Suppose N is sub directly irreducible. First we claim that no non-zero idempotent of N is a zero- divisor. Let J be the set of all non-zero idempotent which are zero-divisors and let β‰  βˆ… . Let 𝐼 = β‹‚ (0 ∢ 𝑒)π‘’βˆˆπ½ . Since N is subdirectly irreducible, 𝐼 β‰  βˆ…. Let π‘Ž ∈ 𝐼 βˆ’ {0}. Thus π‘Žπ‘’ = 0 for all e in J ……….…. (16) This β‡’ 𝑓(π‘Ž)π‘Žπ‘’ = 0 β‡’ 𝑒𝑓(π‘Ž)π‘Ž = 0 [using K (2)] β‡’ 𝑓(π‘Ž) π‘Ž ∈ 𝐽 . From equation (16) we get π‘Žπ‘“(π‘Ž)π‘Ž = 0 β‡’ π‘Ž = 0. … … … … . . . (17). This contradiction implies that no non-zero idempotent of N is a zero-divisor. We shall now prove that N has no non-trivial N-Subgroups. Let M be any N-subgroup of N such that 𝑀 β‰  {0} and let π‘₯(β‰  0) ∈ 𝑀. Let N be a 𝛿1near ring. Then for all x,y in N, π‘₯𝑁𝑦 = 𝑁π‘₯2𝑦2 .Putting π‘₯ = 1, We get π‘₯𝑁. 1 = 𝑁π‘₯2. 1 for all x in N. β‡’ 𝑁𝑦 = 𝑁𝑦2 for all y in N. For any 𝑛 ∈ 𝑁 , there exists 𝑛1 in N such that 𝑛𝑦 = 𝑛1𝑦2 β‡’ (𝑛 βˆ’ 𝑛1𝑦)𝑦 = 0 β‡’ (𝑛 βˆ’ 𝑛1𝑦)𝑓(𝑦)𝑦 = 0 β‡’ 𝑛 βˆ’ 𝑛1𝑦 = 0[by equation 17]β‡’ 𝑛 = 𝑛1𝑦 ∈ 𝑁𝑀 βŠ† 𝑀. Therefore 𝑁 βŠ† 𝑀 i.e. 𝑀 = 𝑁. Thus N has no non-trivial N-subgroups. Clearly for 𝑛 ∈ 𝑁 βˆ’ {0}, Nn is an N-subgroup of N. Consequently 𝑁𝑛 = 𝑁 for all 𝑛 ∈ 𝑁 βˆ’ {0}……..…. (18) .Also, it is clear that 𝑁𝑑 β‰  {0} [since E βŠ† C(N)]βŠ† 𝑁𝑑]. This and equation (18) guarantee that N is a near-field. [Theorem 8.3, Pilz [3])] Converse is obvious. As an immediate consequence of Theorem 3.1, we have the following: Corollary 3.3.2. N has no non-zero zero-divisors if and only if N is a near-field. We are now in a position to give a structure theorem for N. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 352 https://internationalpubls.com Theorem 3.3.3. N is isomorphic to a sub direct product of near-fields. Proof. From Theorem 3.2.4, N is isomorphic to a sub direct product of sub directly irreducible 𝛿1near- ring, Ni's, say. Obviously the existence of a mate function is preserved under homomorphisms. Hence each Ni admits a mate function. Appealing to Theorem 3.3.1 we get N is isomorphic to a sub direct product of near-fields. Remark 3.3.4. From 8.11 of [3], the additive group of a near-field is abelian. It follows that for any 𝛿1 near-ring N with mate functions, (N, +) is abelian. Proposition 3.3.5. Let N be a Boolean near-ring. Then N is 𝛿1if and only if it is a commutative ring. Proof. We observe that identity function is a mate function for N. Appealing to Theorem 3.2.16 and Remark 3.3.4 we see that when N is a 𝛿1near-ring, 𝑁 = 𝐸 βŠ‚ 𝐢(𝑁) and (𝑁, +) is abelian and hence N is a commutative ring. Conversely, N is Boolean and a commutative ring. Then for all x in N,π‘₯𝑛 = 𝑛π‘₯ for all n in Nβ‡’ π‘₯𝑛𝑦 = 𝑛π‘₯𝑦 for all y in N,β‡’ π‘₯𝑁𝑦 = 𝑁π‘₯2𝑦2Hence the result. Proposition 3.3.6. If N is distributively generated and has no non-zero zero-divisors then N is a division ring. Proof. Corollary 3.3.2 guarantees that N is a near-field. Also (N, +) is abelian [by Remark 3.3.4)] Since N is distributively generated, we see that N is a ring [from Theorem 6.6(c) of Pilz [3]] and hence the result. References [1] J.R. Clay, The near-Ring on groups of low order, Math. Z. 104 (1968), 364-371. [2] N.H. McCoy, The Theory of Ring, MacMillan & Co., 1970. [3] G. Pilz, Near-Ring, North Holland/American Elsevier, Amsterdam, 1983. [4] S. Suryanarayanan and N. Ganesan, Stable and Pseudo stable near-ring, Indian J. Pure and Appl. Math 19 (12) (December, 1988), 1206-1216. [5] S. Suryanarayanan, Near-Ring with P3-mate functions, Bull. Malaysian Math. Soc. (Second Series) 19 (1996), 17-24.