On Various Types of Ideals of Gamma Rings and the Corresponding Operator Rings

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Ranu Paul Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 5, Issue 8, (Part -4) August 2015, pp.95-98

RESEARCH ARTICLE OPEN ACCESS

Ranu Paul Department of Mathematics Pandu College, Guwahati 781012 Assam State, India

Abstract The prime objective of this paper is to prove some deep results on various types of Gamma ideals. The characteristics of various types of Gamma ideals viz. prime/maximal/minimal/nilpotent/primary/semi-prime ideals of a Gamma-ring are shown to be maintained in the corresponding right (left) operator rings of the Gamma-rings. The converse problems are also investigated with some good outcomes. Further it is shown that the projective product of two Gamma-rings cannot be simple. AMS Mathematics subject classification: Primary 16A21, 16A48, 16A78 Keywords: Ideals/Simple Gamma-ring/Projective Product of Gamma-rings

I. Introduction left) ideal if the only right (or left) ideal of 푋 Ideals are the backbone of the Gamma-ring contained in 퐼 are 0 and 퐼 itself. theory. Nobusawa developed the notion of a Gamma- ring which is more general than a ring [3]. He Definition 2.5: A nonzero ideal 퐼 of a gamma ring obtained the analogue of the Wedderburn theorem for (푋, ) such that 퐼 ≠ 푋 is said to be maximal ideal, simple Gamma-ring with minimum condition on one- if there exists no proper ideal of 푋 containing 퐼. sided ideals. Barnes [6] weakened slightly the defining conditions for a Gamma-ring, introduced the Definition 2.6: An ideal 퐼 of a gamma ring (푋, ) is notion of prime ideals, primary ideals and radical for said to be primary if it satisfies, a Gamma-ring, and obtained analogues of the 푎훾푏 ⊆ 퐼, 푎 ⊈ 퐼 => 푏 ⊆ 퐽 ∀ 푎, 푏 ∈ 푋 푎푛푑 훾 ∈ . classical Noether-Lasker theorems concerning 푛−1 primary representations of ideals for Gamma-rings. Where 퐽 = {푥 ∈ 푋: 푥훾 푥 ∈ 퐼 푓표푟 푠표푚푒 푛 ∈ Many prominent mathematicians have extended 푁 푎푛푑 훾 ∈ } 푛−1 fruitfully many significant technical results on ideals and 푥훾 푥 = 푥 푤ℎ푒푛 푛 = 1. of general rings to those of Gamma-rings [1,2,4,7,9]. Definition 2.7: An ideal 퐼 of a gamma ring (푋, ) is II. Basic Concepts said to be nilpotent if for some positive integer 푛, 푛 푛 Definition 2.1: A gamma ring (푋,  ) in the sense of 퐼 = 0. Where we denote 퐼 by the set Nabusawa is said to be simple if for any two nonzero 퐼 퐼 … . . 퐼 (all finite sums of the form elements 푥, 푦 ∈ 푋, there exist 훾 ∈ such that 푥1훾1푥2훾2 … . . 훾푛−1푥푛 with 푥푖 ∈ 퐼 and 훾푖 ∈ ). 푥훾푦 ≠ 0.

Definition 2.8: Let 푋1, 1 and 푋2, 2 be two Definition 2.2: If 퐼 is an additive subgroup of a gamma rings. Let 푋 = 푋1 × 푋2 and gamma ring (푋, ) and 푋 퐼 ⊆ 퐼 (or 퐼 푋⊆I), then ┌ = × . Then defining addition and 퐼 is called a left (or right) gamma ideal of 푋. If 퐼 is 1 2 multiplication on 푋 and by, both left and right gamma ideal then it is said to be a 푥1, 푥2 + 푦1, 푦2 = (푥1 + 푦1, 푥2 + 푦2) , gamma ideal of (푋, ) or simply an ideal. 훼 , 훼 + 훽 , 훽 = (훼 + 훽 , 훼 + 훽 ) 1 2 1 2 1 1 2 2 and 푥1, 푥2 훼1, 훼2 푦1, 푦2 = (푥1훼1푦1, 푥2훼2푦2) Definition 2.3: An ideal 퐼 of a gamma ring (푋, ) is for every 푥1, 푥2 , 푦1, 푦2 ∈ 푋 and said to be prime if for any two ideals 퐴 and 퐵 of 푋, 훼1, 훼2 , 훽1, 훽2 ∈ , 퐴 퐵 ⊆ 퐼 => 퐴 ⊆ 퐼 표푟 퐵 ⊆ 퐼. 퐼 is said to be semi- (푋, ) is a gamma ring. We call this gamma ring as prime if for any ideal 푈 of 푋, 푈 푈 ⊆ 퐼 => 푈 ⊆ 퐼. the Projective product of gamma rings.

Definition 2.4: A nonzero right (or left) ideal 퐼 of a gamma ring (푋, ) is said to be a minimal right (or

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III. Main Results Let 푥 ∈ 푋, 푎 ∈ 퐽 be any two elements. Then for any Theorem 3.1: The Projective product of two gamma ∈ , 훾, 푥 ∈ 푅, [훾, 푎] ∈ 푃. Since 푃 is a left ideal of 푅, so, 훾, 푥 훾, 푎 ∈ 푃 => rings 푋1,  1 and 푋2, 2 can never be a simple gamma ring. 훾, 푥훾푎 ∈ 푃 => 푥훾푎 ∈ 퐽 So 퐽 is a left ideal of 푋 and hence the result. Proof: Let (푋, ) be the projective product of the This result can similarly be proved for right ideals also. Thus every ideal in a gamma ring defines gamma rings 푋 , and 푋 , , 1 1 2 2 an ideal in the right operator ring. Let 푥 = 푥1, 0 , 푦 = 0, 푦2 ∈ 푋 be two nonzero elements. Then 푥 ≠ 0, 푦 ≠ 0 1 2 Theorem 3.3: Every prime ideal of (푋, ) defines a Then for any 훼 = 훼1, 훼2 ∈ , we get prime ideal of the right operator ring 푅 and 푥훼푦 = 푥1, 0 훼1, 훼2 0, 푦2 = 푥1훼10, 0훼2푦2 conversely. = 0,0 = 0 Thus for these nonzero 푥, 푦 ∈ 푋, there does not exist Proof: Let 퐼 be a prime ideal of a gamma ring any 훼 ∈ such that 푥훼푦 ≠ 0. (푋, ). We define a subset 푅′ of 푅 by, Thus (푋, ) is not a simple gamma ring and hence ′ ′ 푅 = { 푖 훾푖 , 푥푖 : 훾푖 ∈ , 푥푖 ∈ 퐼}. We show 푅 is a the result. prime ideal of 푅. If 푅 and 퐿 are the right and left operator rings By Result 3.2, 푅′ is an ideal of the right operator ring respectively of the gamma ring (푋, ), then the 푅. We just need to show the prime part. For this let, forms of 푅 and 퐿 are 푥푦 ∈ 푅′, where 푥, 푦 ∈ 푅. 푅 = { 푖 훾푖 , 푥푖 : 훾푖 ∈ , 푥푖 ∈ 푋} and 퐿 = Then 푥 = 푖[훼푖 , 푥푖 ], 푦 = 푗 [훽푗 , 푦푗 ] where 훼푖 , 훽푗 ∈

{ 푖 푥푖 , 훾푖 : 훾푖 ∈ , 푥푖 ∈ 푋} and 푥푖 , 푦푗 ∈ 퐼 Then defining a multiplication in 푅 by, ′ Now, 푥푦 = 푖[훼푖 , 푥푖 ] 푗 [훽푗 , 푦푗 ] ∈ 푅 푖[훼푖 , 푥푖 ] 푗 [훽푗 , 푦푗 ] = 푖,푗 [훼푖, 푥푖 훽푗 푦푗 ], it can be ′ => 푥푦 = 푖,푗 [훼푖 , 푥푖 훽푗 푦푗 ] ∈ 푅 verified that 푅 forms a ring with ordinary addition of => 푥 훽 푦 ∈ 퐼 푓표푟 푖, 푗 endomorphisms and the above defined multiplication. 푖 푗 푗 => 푥 ∈ 퐼 표푟 푦 ∈ 퐼 for 푖, 푗 [Since 퐼 is a prime Similar verification can also be done on the left 푖 푗 operator ring 퐿 with a defined multiplication. In this ideal of 푋] ′ paper we shall discuss various types of ideals in a If 푥푖 ∈ 퐼 푓표푟 푖, then 푖[훼푖 , 푥푖 ] ∈ 푅 for 훼푖 ∈ ′ gamma ring (푋, ) and their corresponding ideals in which implies that 푥 ∈ 푅 . ′ the right operator ring 푅. Again If 푦푗 ∈ 퐼 푓표푟 푗, then 푗 [훽푗 , 푦푗 ] ∈ 푅 for ′ 훽푗 ∈ which implies that 푦 ∈ 푅 . Theorem 3.2: Every left (or right) ideal of (푋, ) Thus, 푥푦 ∈ 푅′ implies 푥 ∈ 푅′ or 푦 ∈ 푅′. So 푅′ is a defines a left (or right) ideal of the right operator ring prime ideal of the right operator ring 푅. 푅 and conversely. Conversely, let 푃 be a prime ideal of 푅. Then 푃 will be of the form

Proof: Let 퐼 be a left ideal of a gamma ring (푋, ). 푃 = { 푖 훾푖 , 푎푖 : 훾푖 ∈ , 푎푖 ∈ 퐽 ⊆ 푋}. We show 퐽 is a We define a subset 푅′ of 푅 by, prime ideal of 푋. ′ ′ Let 푎, 푏 ∈ 푋 be any two elements such that 푎훾푏 ∈ 퐽 푅 = { 푖 훾푖 , 푥푖 : 훾푖 ∈ , 푥푖 ∈ 퐼}. We show 푅 is a left ideal of 푅. for 훾 ∈ . ′ For this let 푥 = 푖[훾푖 , 푥푖 ] ∈ 푅 푎푛푑 푟 = 푗 [훼푗 , 푎푗 ] ∈ Since 푎, 푏 ∈ 푋 and 훾 ∈ , so 훾, 푎 , [훾, 푏] ∈ 푅

푅 be any elements, where 훾푖 , 훼푗 ∈ ; 푥푖 ∈ 퐼; 푎푗 ∈ Again, 푎훾푏 ∈ 퐽 => 훾, 푎훾푏 ∈ 푃 푋 for 푖, 푗. => 훾, 푎 [훾, 푏] ∈ 푃 => 훾, 푎 ∈ 푃 표푟 [훾, 푏] ∈ 푃 [Since Now, 푟푥 = 푗 [훼푗 , 푎푗 ] 푖[훾푖 , 푥푖 ] = 푖,푗 [훼푗 , 푎푗 훾푖 푥푖 ] 푃 is a prime ideal of 푅] Since 푥 ∈ 퐼; 푎 ∈ 푋; 훾 ∈ for 푖, 푗 and 퐼 is a left 푖 푗 푖 If 훾, 푎 ∈ 푃 then 푎 ∈ 퐽 and if 훾, 푏 ∈ 푃 then 푏 ∈ 퐽. ideal of 푋, so ′ ′ Thus 푎훾푏 ∈ 퐽 => 푎 ∈ 퐽 or 푏 ∈ 퐽. So 퐽 is a prime 푎푗 훾푖 푥푖 ∈ 퐼 => 푖,푗 [훼푗 , 푎푗 훾푖 푥푖 ] ∈ 푅 => 푟푥 ∈ 푅 for ideal of 푋 and hence the result. ′ all 푟 ∈ 푅 푎푛푑 푥 ∈ 푅 ′ So 푅 is a left ideal of the right operator ring 푅. Theorem 3.4: Every minimal ideal of (푋, ) defines a minimal ideal of the right operator ring 푅 Conversely, let 푃 be a left ideal of 푅. Then 푃 will be and conversely. of the form 푃 = { 푖 훼푖 , 푥푖 : 훼푖 ∈ , 푥푖 ∈ 퐽 ⊆ 푋}. We show 퐽 is a Proof: Let 퐼 be a minimal ideal of a gamma ring left ideal of 푋. (푋, ). We define a subset 푅′ of 푅 by,

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′ ′ 푅 = { 푖 훾푖 , 푥푖 : 훾푖 ∈  , 푥푖 ∈ 퐼}. We show 푅 is a Thus we get a proper ideal 퐽 of 푋 containing 퐼, minimal ideal of 푅. which contradicts that 퐼 is a maximal idal of 푋. So 푅′ By Result 3.2, 푅′ is an ideal of the right operator ring is a maximal ideal of the right operator ring 푅. 푅. We just need to show the minimal part. Conversely, let 푀 = { 푖 훾푖 , 푥푖 : 훾푖 ∈ , 푥푖 ∈ 퐴} Suppose 푅′ is not minimal. Then there exists an ideal be a maximal ideal of 푅. Then obviously 퐴 is an ideal 퐴 of 푅 in between 0 and 푅′ i.e 퐴 ≠ 0, 퐴 ⊆ 푅′ but of 푋. We show 퐴 is maximal. 퐴 ≠ 푅′. Since 푀 is maximal, so 푀 is nonzero and there does not exist any proper ideal of 푅 containing 푀. Let 퐴 = { 푗 훼푗 , 푦푗 : 훼푗 ∈ , 푦푗 ∈ 퐽 ⊆ 푋}. Since 퐴 is an ideal of 푅 so by result 3.2, 퐽 is also an ideal of 푋. Since 푀 is nonzero so obviously 퐴 is also nonzero. Since 퐴 ≠ 푅′, so there exists an element 푥 = On the contrary, let 퐴 be not maximal. Then ∃ a ′ proper ideal 퐵 of 푋 containing 퐴 i.e 퐴 ⊆ 퐵 ⊊ 푋. 푖[훾푖 , 푥푖 ] ∈ 푅 but 푥 ⊈ 퐴. We construct a subset 푁 = { 훼 , 푦 : 훼 ∈ , 푦 ∈ => 푥푖 ∈ 퐼 but 푥푖 ∉ 퐽 => 퐽 ≠ 퐼. 푗 푗 푗 푗 푗 Again since 퐴 ≠ 푅′ , so obviously 퐽 ⊆ 퐼. Also 퐴 ≠ 퐵} of 푅. Since 퐵⊊푋=>푁⊊푅. 0, so 퐽 ≠ 0. Let 푚 = 푖[훾푖 , 푥푖 ] ∈ 푀 be any element. Thus there exists an ideal 퐽 of 푋 in between 0 and 퐼, Then 훾푖 ∈ and 푥푖 ∈ 퐴 for 푖 which contradicts the fact that 퐼 is a minimal ideal of Since 퐴 ⊆ 퐵 so 푥푖 ∈ 퐴 => 푥푖 ∈ 퐵 => 푖[훾푖 , 푥푖 ] ∈ 푋, i.e our supposition was wrong. So 푅′ is a minimal 푁 => 푚 ∈ 푁 ideal of 푅. So, 푀 ⊆ 푁. Thus we found a proper ideal 푁 of 푅 containing 푀, which contradicts that 푀 is a maximal Conversely, let 푃 = { 푖 훾푖, 푥푖 : 훾푖 ∈ , 푥푖 ∈ 퐼} be a minimal ideal of 푅. Then I is an ideal of 푋. ideal of 푅. So 퐴 is maximal. Hence the result. We show 퐼 is minimal. Suppose not. Then ∃ an ideal 퐽 of 푋 in between 0 and 퐼 i.e 0 ⊊ 퐽 ⊊ 퐼. Theorem 3.6: Every nilpotent ideal of (푋, )

We define a subset 푄 of 푅 by 푄 = { 푗 훼푗 , 푦푗 : 훼푗 ∈ defines a nilpotent ideal of the right operator ring 푅 , 푦푗∈퐽}. Then 푄 is an ideal of 푅. and conversely.

Since 퐽 ≠ 퐼 and 퐽 ⊆ 퐼, so there exists an element Proof: Let 퐼 be a nilpotent ideal of a gamma ring 푥 ∈ 퐼 but 푥 ∉ 퐽. (푋, ). We define a subset 푃 of 푅 by, Then for any 훾 ∈ , 훾, 푥 ∈ 푃 푏푢푡 훾, 푥 ∉ 푄 => 푃 ≠ 푄. 푃 = { 푖 훾푖 , 푥푖 : 훾푖 ∈ , 푥푖 ∈ 퐼}, which is an ideal of Again since 퐽 ⊆ 퐼 => 푄 ⊆ 푃 and 퐽 ≠ 0 => 푄 ≠ 0. the right operator ring 푅. We show 푃 is a nilpotent Thus there exists an ideal 푄 of 푅 which lies in ideal of 푅. between 0 and 푃, which contradicts that 푃 is a Since 퐼 is nilpotent, so there exists a positive integer 푛 minimal ideal of 푅. Thus 퐼 is a minimal ideal of 푋 푛 such that 퐼 = 0 i..e 퐼 퐼 … . . 퐼 = 0. and hence the result. i.e all elements of the form 푥1훾1푥2훾2 … . . 훾푛−1푥푛 are zero, where 푥푖 ∈ 퐼 and 훾푖 ∈ . ….(1) Theorem 3.5: Every maximal ideal of (푋, ) Let 푥 ∈ 푃푛 be any element. Then 푥 will be of the defines a maximal ideal of the right operator ring 푅 form, and conversely. 푥 = 푎1푎2 … . . 푎푛 where 푎푗 = 푖[훾푖푗 , 푥푖푗 ] ∈ 푃.

Then 푥 = 푖[훾푖1 , 푥푖1 ] 푖[훾푖2 , 푥푖2 ] … . . 푖 훾푖푛 , 푥푖푛 Proof: Let 퐼 be a maximal ideal of a gamma ring = 푖[훾푖1 , 푥푖1 훾푖2 푥푖2 … . . 훾푖푛 푥푖푛 ] (푋, ). We define a subset 푅′ of 푅 by, = 푖[훾푖1 , 0] [Using (1)] ′ ′ 푅 = { 푖 훾푖 , 푥푖 : 훾푖 ∈ , 푥푖 ∈ 퐼}. We show 푅 is a = 0 maximal ideal of 푅. Since 퐼 is maximal so 퐼 is So, 푥 ∈ 푃푛 => 푥 = 0 which implies 푃푛 = 0. So 푃 is nonzero and hence 푅′ is also nonzero. a nilpotent ideal of 푅. By Result 3.2, 푅′ is an ideal of the right operator ring Conversely, let 푃 = { 푖 훾푖, 푥푖 : 훾푖 ∈ , 푥푖 ∈ 퐼} 푅. We just need to show the maximal part. be a nilpotent ideal of 푅 i.e there exists a positive ′ On the contrary, if possible let 푅 be not maximal. integer 푛 such that 푃푛 = 0. Then 퐼 is an ideal of 푋. ′ Then there exists a proper ideal 푃 of 푅 containing 푅 We show 퐼푛 = 0. ′ 푛 i.e 푅 ⊆ 푃 ⊆ 푅. Let 푃 = { 푗 훼푗 , 푦푗 : 훼푗 ∈ , 푦푗 ∈ On the contrary, if possible let 퐼 ≠ 0. So there exists 퐽⊆푋}. Then 퐽 is an ideal of 푋. nonzero elements in 퐼푛 . Let 푥 ∈ 퐼 be any element. Let 푡 ∈ 퐼푛 be a nonzero element. => [훾, 푥] ∈ 푅′ for all 훾 ∈ => [훾, 푥] ∈ 푃 for all Then 푡 = 푡1훾1푡2훾2 … . . 훾푛−1푡푛 where 푡푖 ∈ 퐼 and 훾 ∈ => 푥 ∈ 퐽 => 퐼 ⊆ 퐽 훾푖 ∈ and 푡푖 ≠ 0, 훾푖 ≠ 0. Again since 푃 is a proper ideal of 푅 so 퐽 is also a Since 푡푖 ∈ 퐼 => [훾푖−1, 푡푖 ] ∈ 푃 푛 proper ideal of 푋. So, 훾푛 , 푡1 훾1, 푡2 훾2, 푡3 … . . [훾푛−1, 푡푛 ] ∈ 푃

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푛 ′ => [훾푛 , 푡1훾1푡2훾2푡3 … . . 훾푛−1푡푛 ] ∈ 푃 => [훾푛 , 푡] ∈ 푅 = { 푖 훾푖 , 푥푖 : 훾푖 ∈ , 푥푖 ∈ 퐼} is an ideal of the 푃푛=0=>훾푛,푡=0, which is a contradiction because right operator ring 푅. We show 푅′ is a semi-prime 훾푛 ≠ 0 푎푛푑 푡 ≠ 0. ideal of 푅. So, 퐼푛 = 0, i.e 퐼 is nilpotent ideal of 푋 and hence the 2 ′ 푥 = 푖[훾푖 , 푥푖 ] ∈ 푅 be any element such that 푥 ∈ 푅 result. ′ => [훾푖 , 푥푖 ] [훾푖, 푥푖 ] ∈ 푅 Theorem 3.7: Every primary ideal of (푋,  ) defines 푖 푖 ′ a primary ideal of the right operator ring 푅 and => [훾푖 , 푥푖 훾푖 푥푖 ] ∈ 푅 conversely. 푖 => 푥푖 훾푖 푥푖 ∈ 퐼 푓표푟 푎푙푙 푖 Proof: Let 퐼 be a primary ideal of a gamma ring => 푥푖 ∈ 퐼 푓표푟 푎푙푙 푖 [Since 퐼 is semi-prime] ′ (푋, ). Then, => [훾푖 , 푥푖 ] ∈ 푅 ′ 푖 푅 = { 푖 훾푖 , 푥푖 : 훾푖 ∈ , 푥푖 ∈ 퐼} is an ideal of the ′ ′ => 푥 ∈ 푅 right operator ring 푅. We show 푅 is a primary ideal 2 ′ ′ ′ of 푅. Thus we get, 푥 ∈ 푅 => 푥 ∈ 푅 . So 푅 is semi- prime. Let 푥 = 푖[훼푖 , 푥푖 ], 푦 = 푗 [훽푗 , 푦푗 ] ∈ 푅 be any two elements such that 푥푦 ∈ 푅′ Conversely, let 푃 = { 푖 훾푖, 푥푖 : 훾푖 ∈ , 푥푖 ∈ 퐴} be a => [훼 , 푥 ] [훽 , 푦 ] ∈ 푅′ => [훼 , 푥 훽 푦 ] ∈ semi-prime ideal of 푅. Then 퐴 is an ideal of 푋. We 푖 푖 푖 푗 푗 푗 푖,푗 푖 푖 푗 푗 show 퐴 is semi-prime. 푅′=>푥푖훽푗푦푗∈퐼 for 푖,푗 Let 푎 ∈ 푋 be any element such that 푎훾푎 ∈ 퐴 for Since 퐼 is primary, so either 푥 ∈ 퐼 or (푦 훽 )푛−1푦 ∈ 푖 푗 푗 푗 훾 ∈ 퐼 for some 푛 ∈ 푁. => 훾, 푎훾푎 ∈ 푃 => 훾, 푎 훾, 푎 ∈ 푃 => 훾, 푎 2 ∈ If 푥 ∈ 퐼 then [훼 , 푥 ] ∈ 푅′ => 푥 ∈ 푅′ 푖 푖 푖 푖 푃 => 훾, 푎 ∈ 푃 => 푎 ∈ 퐴. And, if (푦 훽 )푛−1푦 ∈ 퐼 then [훽 , (푦 훽 )푛−1푦 ] ∈ 푗 푗 푗 푗 푗 푗 푗 푗 So 퐴 is semi-prime and hence the result. 푅′ ′ => [훽푗 , 푦푗 훽푗 푦푗 훽푗 … . . 훽푗 푦푗 ] ∈ 푅 References 푗 [1] G.L. Booth, Radicals in categories of gamma rings, Math. Japan 35, No. 5(1990), => 훽 , 푦 훽 , 푦 … . . [훽 , 푦 ] ∈ 푅′ 푗 푗 푗 푗 푗 푗 887-895. 푗 푗 푗 [2] J. Luh, On primitive 푛 ′ 푛 ′ -rings with minimal => ( [훽푗 , 푦푗 ]) ∈ 푅 => 푦 ∈ 푅 one sided ideals, Osaka J. Math. 5(1968), 푗 165-173. ′ ′ 푛 ′ ′ So, 푥푦 ∈ 푅 => 푥 ∈ 푅 표푟 푦 ∈ 푅 . Thus 푅 is a [3] N. Nobusawa, On a Generalisation of the primary ideal of 푅. Ring Theory, Osaka J. Math., 1(1964), 81- Conversely, let 푃 = { 푖 훾푖, 푥푖 : 훾푖 ∈ , 푥푖 ∈ 퐴} be a 89. primary ideal of 푅. Then 퐴 is an ideal of 푋. We show [4] T.S. Ravisankar and U.S. Shukla, Structure 퐴 is primary. of ┌-rings, Pacific J. Math. 80, No.2(1979), For this let 푎, 푏 ∈ 푋 be two elements such that 537-559. 푎훾푏 ∈ 퐴, 훾 ∈ [6] W.E. Barnes, On the -rings of Nobusawa, Then, 훾, 푎훾푏 ∈ 푃 => 훾, 푎 훾, 푏 ∈ 푃 Pacific Journal of Math. 18, No.3(1966), Since 푃 is primary ideal of 푅, so 훾, 푎 ∈ 푃 or 411-422. 훾, 푏 푛 ∈ 푃 [7] W.E. Coppage and J. Luh, Radicals of If 훾, 푎 ∈ 푃 then 푎 ∈ 퐴 gamma rings, J. Math. Soc. Japan 23, And, if 훾, 푏 푛 ∈ 푃 then 훾, 푏 훾, 푏 … . . 훾, 푏 ∈ 푃 No.1(1971), 41-52. => 훾, 푏훾푏훾 … . . 훾푏 ∈ 푃 => 훾, (푏훾)푛−1푏 [8] W.G. Lister, Ternary Rings, Transactions of ∈ 푃 => (푏훾)푛−1푏 ∈ 퐴 the American Math. Soc., 154 (1971), 37- Thus 푎훾푏 ∈ 퐴 => 푎 ∈ 퐴 표푟 (푏훾)푛−1푏 ∈ 퐴 55. So 퐴 is a primary ideal of 푋. Hence the result. [9] W.X. Chen, -rings and generalized - rings satisfying the ascending chain condition on left zero divisor ideals, J. Math. Theorem 3.8: Every semi-prime ideal of (푋, ) Res. Exposition 5, No.2(1985), 1-4. defines a semi-prime ideal of the right operator ring

푅 and conversely.

Proof: Let 퐼 be a semi-prime ideal of a gamma ring (푋, ). Then,