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Centralizers on Semiprime Semiring

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Centralizers on Semiprime Semiring 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 푓 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 푚푝푙푒푠 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.
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