IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 12, Issue 3 Ver. IV (May. - Jun. 2016), PP 86-93 www.iosrjournals.org
Centralizers on Semiprime Semiring
D. Mary Florence1, R. Murugesan2, P. Namasivayam3 1 Kanyakumari Community College, Mariagiri – 629153 Tamil Nadu, India 2 Thiruvalluvar College Papanasam - 627425 Tamil Nadu, India 3 The M.D.T Hindu College, Tirunelveli - 627010 Tamil Nadu, India
Abstract: Let 푆 be a 2-torsion free semiprime semiring and 푇: 푆 → 푆 푏푒 an additive mapping. Then we prove that every Jordan left centralizer on 푆 is a left centralizer on 푆. We also prove that every Jordan centralizer of a 2-torsion free Semiprime Semiring is a centralizer. Keywords: Semiring, Semiprime Semiring, left (right) centralizer, Centralizer, Jordan centralizer,
I. Introduction Semiring was introduced by H.S. Vandiver in 1934. Herstein and Neumann investigated Centralizers in rings for a situation which arises in the study of rings with involutions. The study of centralizing mapping on Semiprime ring was established by H.E.Bell and W.S.Martindale. This research has been motivated by the work of Borut Zalar [4] who worked on centralizers of Semiprime rings and proved that Jordan centralizers and centralizers of this rings coincide. Joso Vukman[5],[6], developed some remarkable results using centralizers on prime and Semiprime rings. In [7] Md. Fazlul Hoque and A.C.Paul discussed the notion of Centralizers of Semiprime Gamma Rings. In this paper we introduce the notion of Centralizers of Semiprime Semirings. Throughout, S will represent Semiprime Semiring with center 푍(푆).
II. Preliminaries In this section, we recall some basic definitions and results that are needed for our work.
Definition 2.1 A Semiring is a nonempty set 푆 followed with two binary operation ‘ + ’ and ‘. ’such that (1) (푆, +) is a commutative monoid with identity element ′0’. (2) (푆, . )is a monoid with identity element 1 (3) Multiplication distributes over addition from either side That is 푎, 푏, 푐 푆 , 푎. (푏 + 푐) = 푎. 푏 + 푎. 푐 (푏 + 푐). 푎 = 푏. 푎 + 푐. 푎 Definition 2.2 A Semiring 푆 is said to be prime if 푥푆푦 = 0 ⟹ 푥 = 0 표푟 푦 = 0 푥, 푦푆 Definition 2.3 A Semiring 푆 is said to be semiprime if 푥푆푥 = 0 ⟹ 푥 = 0 푥푆 Definition 2.4 A Semiring 푆 is said to be 2-torsion free if 2푥 = 0 ⟹ 푥 = 0 푥푆 Definition 2.5 A Semiring 푆 is said to be commutative Semiring, if 푥푦 = 푦푥 푥, 푦 푆, then the set 푍(푆) = { 푥 푆, 푥푦 = 푦푥 푦 푆} is called the center of the Semiring 푆. Definition 2.6 For any fixed 푎푆, the mapping 푇(푥) = 푎푥 is a left centralizer and 푇(푥) = 푥푎 is a right centralizer. Definition 2.7 An additive mapping 푇: 푆 → 푆 is a left (right) centralizer 𝑖푓 푇(푥푦) = 푇(푥)푦, (푇(푥푦) = 푥푇(푦)) 푥, 푦푆 A centralizer is an additive mapping which is both left and right centralizer. Definition 2.8 An additive mapping 푇: 푆 → 푆is Jordan left (right) Centralizer if 푇(푥푥) = 푇(푥)푥, (푇(푥푥) = 푥푇(푥)) 푥 푆 Every left centralizer is a Jordan left centralizer, but the converse is not in general true. Definition 2.9 An additive mapping 푇: 푆 → 푆is a Jordan centralizer if 푇(푥푦 + 푦푥) = 푇(푥)푦 + 푦푇(푥). 푥, 푦푆 Every centralizer is a Jordan centralizer, but Jordan centralizer is not in general a centralizer. Definition 2.10 An additive mapping 퐷: 푆 → 푆 is called a derivation if 퐷(푥푦) = 퐷(푥)푦 + 푥퐷(푦) 푥, 푦 푆 and is called a Jordan derivation if 퐷(푥푥) = 퐷(푥)푥 + 푥퐷(푥) 푥 푆.
DOI: 10.9790/5728-1203048693 www.iosrjournals.org 86 | Page Centralizers on Semiprime Semiring
Definition 2.11 If 푆 is a semiring then [푥, 푦] = 푥푦 + 푦′푥 is known as the commutator of 푥 and 푦 The following are the basic commutator identities: [푥푦, 푧] = [푥, 푧]푦 + 푥[푦, 푧] and [푥, 푦푧] = [푥, 푦]푧 + 푦[푥, 푧]. 푥, 푦, 푧푆 According to [11] 푓표푟 푎푙푙 푎, 푏 푆 we have (푎 + 푏)′ = 푎′ + 푏′ (푎푏)′ = 푎′푏 = 푎푏′ 푎′′ = 푎 푎′푏 = (푎′푏)′ = (푎푏)′′ = 푎푏 Also the following implication is valid. 푎 + 푏 = 표 𝑖푚푝푙𝑖푒푠 푎 = 푏′ and 푎 + 푎′ = 0
III. Left Centralizers Of Semiprime Semirings Lemma 3.1[2] Let S be a 2-torsion free Semiprime Semirings , 푎 and 푏 the elements of 푆, then the following are equivalent. (i) 푎푆푏 = 0 (ii) 푏푆푎 = 0 (iii) 푎푆푏 + 푏푆푎 = 0 If one of these conditions are fulfilled, then 푎푏 = 푏푎 = 0
Lemma 3.2 Let 푆 be a semiprime semiring and 퐴: 푆 x 푆 → 푆 biadditive mapping. If 퐴(푥, 푦)푤퐵(푥, 푦) = 0 푥, 푦푆. then 퐴(푥, 푦)푤퐵(푠, 푡) = 0 푥, 푦, 푠, 푡, 푤 푆
Proof By hypothesis 퐴(푥, 푦)푤퐵(푥, 푦) = 0, (1) Replace 푥 푏푦 푥 + 푠 in the above, we get 퐴(푥, 푦)푤퐵(푠, 푦) = 퐴(푠, 푦)푤′퐵(푥, 푦) This implies (퐴(푥, 푦)푤퐵(푠, 푦))푧(퐴(푥, 푦)푤퐵(푠, 푦)) = (퐴(푠, 푦)푤′퐵(푥, 푦))푧 (퐴(푥, 푦)푤퐵(푠, 푦)) = 퐴(푠, 푦)푤′ (퐵(푥, 푦)푧 퐴(푥, 푦)) 푤 퐵(푠, 푦) Using lemms 3.1 and by the semiprimeness of 푆 implies, 퐴(푥, 푦)푤퐵(푠, 푦) = 0 Now replacing 푦 by 푦 + 푡 in the last equation and using similar approach we get the required result. Lemma 3.3 Let 푆 be a Semiprime Semiring and 푎푆 be some fixed element. If 푎[푥, 푦] = 0 for 푥, 푦 푆, then there exists an ideal 푈 of 푆 such that 푎푈 ⊂ 푍(푆) holds.
Proof Let 푥, 푦, 푧 푆 Consider [푧, 푎]푥[푧, 푎] = (푧푎 + 푎′푧)푥[푧, 푎] = 푧푎푥[푧, 푎] + 푎′푧푥[푧, 푎] = 푧푎{[푧, 푥푎] + [푧, 푥]푎′} + 푎′{[푧, 푧푥푎] + [푧, 푧푥]푎′} = 푧푎{[푧, 푥푎] + [푧, 푥]푎′} + 푎′[푧, 푧푥푎] + 푎′ [푧, 푧푥]푎′} = 푧푎{푥[푧, 푎] + [푧, 푥]푎 + [푧, 푥]푎′} + 푎′{[푧, 푧]푥푎 + 푧[푧, 푥]푎 + 푧푥[푧, 푎]} +푎′(푧[푧, 푥] + [푧, 푧]푥)푎′ =0 Since 푆 is semiprime, [푧, 푎] = 0 푧푎 + 푎′푧 = 0.This implies 푎푍(푆) Also, 푧푎푤[푥, 푦] = 푧푤푎[푥, 푦] 푥, 푦, 푧, 푤 푆 =0 for all 푥, 푦, 푧, 푤 푆 By similar arguments we can show that 푧푎푤푍(푆) Thus we obtain 푆푎푆 ⊂ 푍(푆), and it is easy to see that ideal generated by a is central.
Lemma 3.4
DOI: 10.9790/5728-1203048693 www.iosrjournals.org 87 | Page Centralizers on Semiprime Semiring
Let 푆 be a semiprime semiring and let T : 푆 → 푆 be a Jordan left centralizer. Then (a) 푇(푥푦 + 푦푥) = 푇(푥)푦 + 푇(푦)푥 푥, 푦 푆 (b) If 푆 is a 2- torsion free Semiring, then (i) 푇(푥푦푥) = 푇(푥)푦푥 (ii) 푇(푥푦푧 + 푧푦푥) = 푇(푥)푦푧 + 푇(푧)푦푥 푥, 푦, 푧 푆
Proof T is a Jordon left centralizer then 푇(푥푥) = 푇(푥)푥 (2) Replace 푥 푏푦 푥 + 푦 푇[(푥 + 푦)(푥 + 푦)] = [푇(푥 + 푦)](푥 + 푦) 푇(푥푥 + 푥푦 + 푦푥 + 푦푦] = [푇(푥) + 푇(푦)] (푥 + 푦) 푇(푥푥) + 푇(푥푦 + 푦푥) + 푇(푦푦) = 푇(푥)푥 + 푇(푥)푦 + 푇(푦)푥 + 푇(푦)푦 푇(푥)푥 + 푇(푥푦 + 푦푥) + 푇(푦)푦 = 푇(푥)푥 + 푇(푥)푦 + 푇(푦)푥 + 푇(푦)푦 푇(푥푦 + 푦푥) = 푇(푥)푦 + 푇(푦)푥 (3) Hence proved (a) Now replacing 푦 푏푦 푥푦 + 푦푥 in (3) we get 푇[푥(푥푦 + 푦푥) + (푥푦 + 푦푥)푥] = 푇(푥)[푥푦 + 푦푥] + 푇(푥푦 + 푦푥)푥 푇(푥2푦 + 푥푦푥 + 푥푦푥 + 푦푥2) = [푇(푥)푥푦 + 푇(푥)푦푥 + 푇(푥푦 + 푦푥)푥 푇(푥2)푦 + 푇(푦)푥2 + 2푇(푥푦푥) = 푇(푥) 푥푦 + 푇(푥)푦푥 + [푇(푥)푦 + 푇(푦)푥]푥 푇(푥2)푦 + 푇(푦)푥2 + 2푇(푥푦푥) = 푇(푥2)푦 + 푇(푥)푦푥 + 푇(푥)푦푥 + 푇(푦)푥2 2푇(푥푦푥) = 2푇(푥)푦푥
Adding 2푇(푥)푦푥′ on bothsides, we get 2푇(푥푦푥) + 2푇(푥)푦푥′ = 0
Since 푆 is 2-torsion free, 푇(푥푦푥) + 푇(푥)푦푥′ = 0 This implies, 푇(푥푦푥) = 푇(푥)푦푥 (4) Hence proved b(i) If we linearize (4), we get 푇(푥푦푧 + 푧푦푥) = 푇(푥)푦푧 + 푇(푧)푦푥 푥, 푦, 푧 푆 Hence b(ii) is proved.
Theorem 3.1 Let 푆 be a 2- torsion free Semiprime Semiring and 푇: 푆 → 푆 be a Jordan Left centralizer on 푆. Then 푇 is a left centralizer.
Proof 푇 is a Jordan left centralizer then 푇(푥푦 + 푦푥) = 푇(푥)푦 + 푇(푦)푥 (5) First we shall compute 퐽 = 푇(푥푦푧푦푥 + 푦푥푧푥푦)in two different ways Using Lemma 3.4. b(i), we have 퐽 = 푇(푥)푦푧푦푥 + 푇(푦)푥푧푥푦 (6) Using Lemma 3.4. b(ii), we have 퐽 = 푇(푥푦)푧푦푥 + 푇(푦푥)푧푥푦 (7) Comparing (1) and (2) 푇(푥)푦푧푦푥 + 푇(푦)푥푧푥푦 = 푇(푥푦)푧푦푥 + 푇(푦푥)푧푥푦 푇(푦)푥푧푥푦 = 푇(푥푦)푧푦푥 + 푇′(푥)푦푧푦푥 + 푇(푦푥)푧푥푦 푇(푦)푥푧푥푦 = (푇(푥푦) + 푇(푥)푦′)푧푦푥 + 푇(푦푥)푧푥푦 Adding 푇(푦)푥푧푥푦′ on both sides, we get (푇(푥푦) + 푇(푥)푦′)푧푦푥 + 푇(푦푥)푧푥푦 + 푇(푦)푥푧푥푦′ = 0 (푇(푥푦) + 푇(푥)푦′)푧푦푥 + (푇(푦푥) + 푇(푦)푥′)푧푥푦 = 0
Introducing a biadditive mapping 퐵: 푆푋푆 → 푆 by 퐵(푥, 푦) = 푇(푥푦) + 푇(푥)푦′ From the above relation we arrive at 퐵(푥, 푦)푧푦푥 + 퐵(푦, 푥)푧푥푦 = 0 (8) Using 퐵(푦, 푥) = 퐵′(푥, 푦) in the above, we get 퐵(푥, 푦)푧[푥, 푦] = 0
Using Lemma 3.2 we get, 퐵(푥, 푦)푧[푠, 푡] = 0 Using Lemma 3.1 we obtain 퐵(푥, 푦)[푠, 푡] = 0
DOI: 10.9790/5728-1203048693 www.iosrjournals.org 88 | Page Centralizers on Semiprime Semiring
Now fix some 푥, 푦 푆. Write 푏 = 퐵(푥, 푦), 푏푆 푡ℎ푒푛 푏[푠, 푡] = 0, 푓표푟 푠, 푡 푆 . 퐵푦 푙푒푚푚푎 3.3 there exist an ideal 푈 such that 푏 푈 ⊂ 푍(푆) holds. In particular 푏푦, 푦푏 푍(푆), 푦푆, This gives us 푥(푏2푦) = (푏2푦)푥 = (푦푏2)푥 푥푏2푦 = 푦푏2푥 푇(푥푏2푦) = 푇 (푦푏2푥) Linearizing the above 푇(푥푏2푦 + 푦푏2푥) = 푇 (푦푏2푥 + 푥푏2푦) 푇(푥푏2푦 + 푥푏2푦) = 푇 (푦푏2푥 + 푦푏2푥) 2푇(푥푏2푦) = 2푇(푦푏2푥) 푇ℎ𝑖푠 푔𝑖푣푒푠 푢푠 4푇 (푥푏2푦) = 4푇(푦푏2푥) Now we will compute each side of this equality by using (5) and the above notation 4푇(푥푏2푦) = 2푇 [ 푥푏2푦 + 푥푏2푦] = 2푇[푥푏2푦 + 푦푏2푥] = 2푇 [푥푏2푦 + 푏2푦푥] Similarly 4푇(푦푏2푥) = 2푇(푦푏2푥 + 푦푏2푥) = 2푇 (푦푏2푥 + 푥푏2푦) = 2푇 (푦푏2푥 + 푏2푥푦) Therefore we get 2푇(푥푏2푦 + 푏2푦푥) = 2푇 (푦푏2푥 + 푏2푥푦) 2[푇(푥)푏2푦 + 푇(푏2푦)푥] = 2[푇(푦)푏2푥 + 푇(푏2푥)푦] 2푇(푥)푏2푦 + 2푇(푏2푦)푥 = 2푇(푦)푏2푥 + 2푇(푏2푥)푦 2푇(푥)푏2푦 + 푇[푏2푦 + 푏2푦]푥 = 2푇(푦)푏2푥 + 푇[푏2푥 + 푏2푥]푦 2푇(푥)푏2푦 + 푇[푏2푦 + 푦푏2]푥 = 2푇(푦) 푏2푥 + 푇[푏2푥 + 푥푏2]푦 2푇(푥)푏2푦 + [푇(푏2)푦 + 푇(푦)푏2]푥 = 2푇(푦)푏2푥 + [푇(푏2)푥 + 푇(푥)푏2]푦 2푇(푥)푏2푦 + 푇(푏2)푦푥 + 푇(푦)푏2푥 = 2푇(푦)푏2푥 + 푇(푏2)푥푦 + 푇(푥)푏2푦 푇(푥)푏2푦 + 푇(푏)푏푦푥 = 푇(푦)푏2푥 + 푇(푏) 푏푥푦 푇(푥)푏2푦 + 푇(푏)푏푥푦 = 푇(푦)푏2푥 + 푇(푏)푏푥푦 푇(푥)푏2푦 = 푇(푦)푏2푥 (9)
On the other hand we also have 푥(푏2푦) = 푥푏푏푦 푥푏2푦 = 푥푏푦푏 푥푦푏2 = 푥푏푦푏 푇(푥푦푏2 ) = 푇(푥푏푦푏)
Linearizing the above and using (5) and (9), we get
푇(푥푦푏2 + 푦푥푏2) = 푇(푥푏푦푏 + 푦푏푥푏) 푇(푥푦푏2 + 푥푦푏2) = 푇(푥푏푦푏 + 푥푏푦푏) 2푇(푥푦푏2) = 2푇(푥푏푦푏) This gives us 4푇(푥푦푏2) = 4푇(푥푏푦푏) 2푇(푥푦푏2 + 푥푦푏2) = 2푇(푥푏푦푏 + 푥푏푦푏) 2푇(푥푦푏2 + 푏2푥푦) = 2푇(푥푏푦푏 + 푦푏푥푏) 2 푇(푥푦)푏2 + 2 푇(푏2)푥푦 =2 푇(푥푏)푦푏 + 2 푇(푦푏)푥푏 2 푇(푥푦)푏2 + 2 푇(푏)푏푥푦 = 푇(푥푏 + 푥푏)푦푏 + 푇(푦푏 + 푦푏)푥푏 = 푇(푥푏 + 푏푥)푏푦 + 푇(푦푏 + 푏푦)푏푥 = [푇(푥)푏 + 푇(푏)푥]푏푦 + [푇(푦)푏 + 푇(푏)푦]푏푥 = 푇(푥)푏2푦 + 푇(푏)푏푥푦 + 푇(푦)푏2푥 + 푇(푏)푏푥푦 2 푇(푥푦)푏2 = 푇(푥)푏2푦 + 푇(푦)푏2푥 2 푇(푥푦)푏2 = 푇(푥)푏2푦 + 푇(푥)푏2푦 Adding 2푇(푥)푏2푦′ on both sides 2 푇(푥푦)푏2 + 2 푇(푥)푏2푦′=0 Since 푆is 2 – torsion, [푇(푥푦) + 푇(푥)푦′]푏2]=0 푏푏2 = 0, 푏3 = 0, So that, 푏2푆푏2 = 푆푏2푏2
DOI: 10.9790/5728-1203048693 www.iosrjournals.org 89 | Page Centralizers on Semiprime Semiring
= 푆푏푏3 = 0 Since 푆 is Semiprime 푏2 = 0 Also 푏푆푏 = 푆푏푏 = 푆푏2 = 0 Since 푆 is Semiprime, 푏 = 0 T(푥푦) + 푇(푥)푦′ = 0 푇(푥푦) = 푇(푥)푦 Hence T is a left centralizer.
IV. The Centralizers Of Semiprime Semirings Lemma 4.1 Let 푆 be a semiprime semiring. 퐷 𝑖푠 a deveriation of 푆 and 푎 푆 be some fixed element. Then (i) 퐷(푥)퐷(푦) = 0 푥, 푦 푆 ⟹ 퐷 = 0 (ii) 푎푥 + 푥′푎 푍(푆) 푥푆 ⟹ 푎푍(푆)
Proof i) Consider 퐷(푥)푦퐷(푥) = 퐷(푥)[퐷(푦푥) + 퐷(푦)푥′] = 퐷(푥)퐷(푦푥) + 퐷(푥)퐷(푦)푥′ Replace 푦푥 푏푦 푧 푥, 푦, 푧 푆. 퐷(푥)푦퐷(푥) = 퐷(푥)퐷(푧) + 퐷(푥)퐷(푦)푥′ = 0 Since S is Semiprime, 퐷 = 0, hence Proved (i) ii) Define 퐷(푥) = 푎푥 + 푥′푎푍(푆) 퐷(푥푦) = 푎푥푦 + 푥′푦푎 , since 퐷 is a derivation, we get 퐷(푥)푦 + 푥퐷(푦) = 푎푥푦 + 푥′푦푎 Since 퐷(푥)푍(푆), 푥 푆 we have 퐷(푦)푥 = 푥퐷(푦) and also 퐷(푦푧)푥 = 푥퐷(푦푧) [퐷(푦)푧 + 푦퐷(푧)]푥 = 푥[퐷(푦)푧 + 푦퐷(푧)] 퐷(푦)푧푥 + 푦퐷(푧)푥 = 푥퐷(푦)푧 + 푥푦퐷(푧) 퐷(푦)푧푥 = 푥푦퐷(푧) + 푦′퐷(푧)푥 + 푥퐷(푦)푧 = 푥퐷(푦)푧 + 푥퐷(푧)푦+퐷(푧)푦′푥 = 푥퐷(푦)푧+ 퐷(푧)푥푦+퐷(푧)푦′푥 = 푥퐷(푦)푧+ 퐷(푧)[푥, 푦]
Obviously 퐷(푎) = 0, now take 푧 = 푎, we get 퐷(푦)푎푥 = 푥퐷(푦)푎+ 퐷(푎)[푥, 푦] 퐷(푦)푎푥 = 푥퐷(푦)푎 Adding 푥′퐷(푦)푎 on both sides, We get 퐷(푦)푎푥 + 푥′퐷(푦)푎 = 0 퐷(푦)푎푥 + 퐷(푦)푥′푎=0 퐷(푦)퐷(푥) = 0 By (i) 퐷 = 0 퐷(푥) = 0 푎푥 + 푥′푎 = 0 푎푍(푆)
Lemma 4.2 Let 푆 be a semiprime semiring and 푎푆 be some fixed element. If 푇(푥) = 푎푥 + 푥푎 is a Jordan centralizer, then 푎푍(푆).
Proof By hypothesis, 푇 is a Jordan centralizer we have 푇(푥푦 + 푦푥) = 푇(푥)푦 + 푦푇(푥) (10) Given 푇(푥) = 푎푥 + 푥푎 then replace 푥 = 푥푦 + 푦푥 푇(푥푦 + 푦푥) = 푎(푥푦 + 푦푥) + ( 푥푦 + 푦푥)푎 (11) Comparing (10) and (11) 푎(푥푦 + 푦푥) + ( 푥푦 + 푦푥)푎 = 푇(푥)푦 + 푦푇(푥) 푎(푥푦 + 푦푥) + ( 푥푦 + 푦푥)푎 = [푎푥 + 푥푎]푦 + 푦[푎푥 + 푥푎]
DOI: 10.9790/5728-1203048693 www.iosrjournals.org 90 | Page Centralizers on Semiprime Semiring
푎푦푥 + 푥푦푎 = 푥푎푦 + 푦푎푥 adding 푥′푎푦 + 푦푎푥′ on both sides 푎푦푥 + 푥푦푎 + 푥′푎푦 + 푦푎푥′ = 0 (푎푦 + 푦′푎)푥 + 푥′ (푎푦 + 푦′푎) = 0 푎푦 + 푦′푎 푍(푆) by Lemma 4.1, (ii) gives us 푎푍(푆). Hence the proof is complete. Lemma 4.3 퐿푒푡 푆 푏푒 푎 푠푒푚𝑖푝푟𝑖푚푒 푠푒푚𝑖푟𝑖푛푔. 푇ℎ푒푛 푒푣푒푟푦 퐽표푟푑표푛 푐푒푛푡푟푎푙𝑖푧푒푟 표푓 푆 푚푎푝푠 푍(푆) 𝑖푛푡표 푍(푆)
Proof Take any 푐푍(푆)and 푎 = 푇(푐), then we have 2푇(푐푥) = 푇(푐푥 + 푐푥) = 푇(푐푥 + 푥푐) = 푇(푐)푥 + 푥 푇(푐) = 푎푥 + 푥푎
Let푓(푥) = 2푇(푐푥) Replace 푥 = 푥푦 + 푦푥 푓(푥푦 + 푦푥) = 2푇[푐(푥푦 + 푦푥)] = 2푇[푐푥푦 + 푦푐푥] = 2[푇(푐푥)푦 + 푦푇(푐푥)] = 2푇(푐푥)푦 + 푦2푇(푐푥) = 푓(푥)푦 + 푦푓(푥) Hence푓(푥) is a Jordan centralizer. Therefore 푓(푥) = 푎푥 + 푥푎 is a Jordan centralizer. By Lemma 4.2, 푎푍(푆) ⟹ 푇(푐)푍(푆), hence complete the proof.
Lemma 4.4 퐿푒푡 푆 푏푒 푎 푆푒푚𝑖푝푟𝑖푚푒 푆푒푚𝑖푟𝑖푛푔 푎푛푑 푎, 푏푆 푏푒 푡푤표 푓𝑖푥푒푑 푒푙푒푚푒푛푡푠. 퐼푓 푎푥 + 푥′푏 = 0 푥푆, 푡ℎ푒푛 푎 = 푏푍(푆)
Proof By hypothesis 푎푥 + 푥′푏 = 0 , we have 푎푥 = 푥푏 Replace 푥 by 푥푦 푎푥푦+ 푥′푦푏=0 푥[푏, 푦] = 0, 푏, 푥, 푦푆 Hence [푏, 푦]푥[푏, 푦] = 0 Since 푆 is semipime, [푏, 푦] = 0 푏푦 + 푦′푏 = 0 푏푍(푆) We have 푎푥 + 푥′푏 = 0 푎푥 + 푏′푥 = 0 (푎 + 푏′)푥 = 0 hence (푎 + 푏′)푥(푎 + 푏′) = 0 By the semiprimeness of 푆 implies 푎 + 푏′ = 0
푎 = 푏 푍(푆)
Theorem 4.1 Every Jordan Centralizer of 푎 2-torsion 푓푟푒푒 푆푒푚𝑖푝푟𝑖푚푒 푆푒푚𝑖푟𝑖푛푔 𝑖푠 푎 푐푒푛푡푟푎푙𝑖푧푒푟.
Proof Suppose 푇 is a Jordan Centralizer, Then 푇(푥푦 + 푦푥) = 푇(푥)푦 + 푦푇(푥) = 푥푇(푦) + 푇(푦)푥 푇(푥)푦 + 푦푇(푥) = 푥푇(푦) + 푇(푦)푥 , Replace 푦 by 푥푦 + 푦푥, we get 푇(푥)(푥푦 + 푦푥) + (푥푦 + 푦푥)푇(푥) = 푥푇(푥푦 + 푦푥) + 푇(푥푦 + 푦푥)푥 푇(푥)푥푦 + 푇(푥)푦푥 + 푥푦푇(푥) + 푦푥푇(푥) = 푥푇(푥)푦 + 푥푦푇(푥) + 푇(푥)푦푥 + 푦푇(푥)푥 푇(푥)푥푦 = 푥푇(푥)푦 + 푦푇(푥)푥 + 푦푥′푇(푥) 푇(푥)푥푦 = 푥푇(푥)푦+ 푦[푇(푥), 푥]
DOI: 10.9790/5728-1203048693 www.iosrjournals.org 91 | Page Centralizers on Semiprime Semiring
Adding 푥′푇(푥)푦 on both sides; 푇(푥)푥푦 + 푥′푇(푥)푦 = 푦[푇(푥), 푥] [푇(푥), 푥]y=푦[푇(푥), 푥] [푇(푥), 푥] ∈ 푍(푆)
Next our goal is to show that [푇(푥), 푥] = 0 Take any 푐 ∈ 푍(푆). Then 2푇(푐푥) = 푇(푐푥 + 푐푥) = 푇(푐푥 + 푥푐) = 푐푇(푥) + 푇(푥)푐 = 푇(푥)푐 + 푇(푥)푐 = 2푇(푥)푐 퐴푑푑𝑖푛푔 2푇′(푥)푐 on both sides 2푇(푐푥) + 2푇′(푥)푐 = 0, this implies 푇(푐푥) = 푇(푥)푐
By lemma 4.3, T(C)∈ 푍(푆), Hence 푇(푐)푥 = 푥푇(푐) Hence 푇(푐푥) = 푇(푐)푥 = 푇(푥)푐 [푇(푥), 푥]푐 = [푇(푥)푥 + 푥′푇(푥)]푐 = 푇(푥)푥푐 + 푥′푇(푥)푐 = 푇(푥)푐푥 + 푥′푇(푐)푥 = 푇(푐)푥푥 + 푇(푐)푥푥′ = 0 퐻푒푛푐푒 [푇(푥), 푥]푐 [푇(푥), 푥] = 0 Thus [푇(푥), 푥] itself is central element, our goal is achieved 2푇(푥푥) = 푇(푥푥 + 푥푥) = 푇(푥)푥 + 푥푇(푥) = 푇(푥)푥 + 푇(푥)푥 = 2푇(푥)푥 퐴푑푑𝑖푛푔 2푇(푥)푥′ on both sides
푇(푥푥) + 푇(푥)푥′ = 0 Since 푆 𝑖푠 2 푡표푟푠𝑖표푛 푇(푥푥) = 푇(푥)푥 Hence 푇 is Jordan left centralzer. By theorem 3.1, 푇 is a left centralizer. Similarly, we have prove that T is a right centralizer. Therefore T is a centralizer
Example 4.1 Let 푆 be a semiring. Define, 푆1 = {(푥, 푥): 푥 ∈ 푆}. Let the operations of addition and multiplication on 푆1 be defined by (푥1, 푥1) + (푥2, 푥2) = (푥1 + 푥2, 푥1 + 푥2) (푥1, 푥1)(푥2, 푥2) = (푥1푥2, 푥1푥2 ) for every 푥1, 푥2 ∈ 푆. Then it can easily seen that 푆1 is a semiring. Let 푑1: 푆 → 푆is a left centralizing mapping and 푑2: 푆 → 푆 be a right centralizing and commutating mapping . Let 푇: 푆1 → 푆1 be the additive mapping defined by 푇(푥, 푥) = (푑1(푥), 푑2(푥)) Let (푥, 푥) = 푎 ∈ 푆1. We have푇(푎, 푎) = 푇((푥, 푥), (푥, 푥)) = 푇(푥푥, 푥푥) = (푑1(푥푥), 푑2(푥푥)) = (푑1(푥)푥, 푥푑2(푥) = (푑1(푥)푥, 푑2(푥)푥 ] = ((푑1(푥), 푑2(푥))(푥, 푥) = 푇(푥, 푥)(푥, 푥) = 푇(푎)푎
Hence 푇 is a Jordan left centralizer, which is not a centralizer.
Example 4.2 Let 푆 be a semiring. Define 푆1 = {(푥, 푥): 푥 ∈ 푆} Let the operations of addition and multiplication on 푆1 be defined by
DOI: 10.9790/5728-1203048693 www.iosrjournals.org 92 | Page Centralizers on Semiprime Semiring
(푥1, 푥1) + (푥2, 푥2) = (푥1 + 푥2, 푥1 + 푥2 ) (푥1, 푥1)(푥2, 푥2) = (푥1푥2, 푥1푥2 ) for every 푥1,푥2 ∈ 푆 Then it can easily seen that 푆1 is a semiring. Let푑1: 푆 → 푆is a left centralizing mapping and 푑2: 푆 → 푆be a right centralizing and commutating mapping. Let 푇: 푆1 → 푆1be the additive mapping defined by 푇(푥, 푥) = (푑1(푥), 푑2(푥)) Let (푥, 푥) = 푎 and (푦, 푦) = 푏
푇(푎푏 + 푏푎) = 푇[(푥, 푥)(푦, 푦) + (푦, 푦)(푥, 푥)] = 푇(푥푦, 푥푦) + (푦푥, 푦푥) = 푇[(푥푦 + 푦푥), (푥푦 + 푦푥)] = (푑1(푥푦 + 푦푥), 푑2(푥푦 + 푦푥)) = [(푑1(푥)푦+푑1(푦)푥), (푥푑2(푦) + 푦푑2(푥))] = [(푑1(푥)푦 + 푑1(푦)푥), (푑2(푦)푥 + 푑2(푥)푦] = [푑1(푥)푦 + 푑1(푦)푥, (푑2(푥)푦 + 푑2(푦)푥] = (푑1(푥)푦, 푑2(푥)푦) + (푑1(푦)푥푑2(푦)푥) = (푑1(푥), 푑2(푥))(푦, 푦) + (푑1(푦)푑2(푦))(푥, 푥) = 푇(푥, 푥)(푦, 푦) + 푇(푦, 푦)(푥, 푥) = 푇(푎)푏 + 푇(푏)푎 Hence T is Jordan Centralizer Which is not a centralizer.
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