Advances in Algebra ISSN 0973-6964 Volume 11, Number 1 (2018), pp. 19-28 © Research India Publications http://www.ripublication.com On Right Centralizers of Semiprime Semirings D.Mary Florence Department of Mathematics, Kanyakumari Community College, Mariagiri – 629153, Tamil Nadu, India.E-mail: [email protected] R. Murugesan, Department of Mathematics, Thiruvalluvar College, Papanasam – 627425, Tamil Nadu, India.E-mail: [email protected] P. Namasivayam Department of Mathematics, The M.D.T Hindu College, Tirunelveli – 627010, Tamil Nadu, India.E-mail: [email protected] Abstract Let 푆 be a semiprime semiring with centre 푍. A mapping ℎ: 푆 → 푆 is called centralizing on 푆 if [푥 , ℎ(푥 )] 휖 푍(푆) for all 푥휖푆. We show that every centralizing right centralizer of a semiprime semiring is a centralizer. We also show that if 푇 and 퐻 are Right centralizers of a semiprime semiring 푆 satisfying 푥푇(푥) + 퐻(푥)푥 휖 푍(푆) for all 푥 휖 푆. Then both 푇 and 퐻 are centralizers of 푆. AMS Subject Classification : 16Y60 Keywords: semiring, semiprime semiring, left(right) centralizer, centralizer, commuting mappings, centralizing mappings, Jordan subring.. 1. INTRODUCTION AND PRELIMINARIES Without giving the formal definition of semiring, semirings first appear implicitly in Richard Dedekind’s supplement 11. The algebraic structure of semiring was introduced by H.S. Vandiver in 1934. This system consisting of a nonempty set 푅 20 D.Mary Florence, R. Murugesan and P. Namasivayam with two binary operations, addition (usually denoted by “+”) and multiplication (usually denoted by “.”) such that the multiplication is distributive over addition. A natural example of semiring which is not a ring is set of Natural numbers 푁, under usual addition and multiplication of numbers. Numerous researchers made good work on centralizers of rings. Recently many authors established the study of semirings and semiprime rings. Herstein and Neuman investigated centralizers in Rings which arises in the study of rings with involution. Zalar [1] worked on centralizers of semiprime rings. M.F.Hoque and A.C.Paul [6,7] studied on centralizers of semiprime Γ-rings . Postner initiated the study of centralizing mappings in Rings. M.Breser and J. Vukman [2,3] established very striking results on centralizing mappings in prime rings. H.E. Bell, W.S.Martindale III [11] proved some remarkable results on centralizing mappings of semiprime Rings . Golan [9] discussed the notion of semirings and their applications. But not gave more brief explanations about centralizers of semirings. Chandramouleeswaran and Tiruveni [4] studied derivations on semiring. M.S. Sammam and M.A.Chaudhry [12] proved that, If two left centralizers 푇 and 푆 of a semiprime ring R satisfying 푇(푥)푥 + 푥푆(푥) 휖 푍(푅), then both 푇 and 푆 are centralizers. K.K.Day and A.C.Paul [8] made the general frame work on left centralizers of semiprime Gamma ring. The purpose of this paper is to propose the notion of Right centralizers of semiprime semiring and proved some simple results. The results in this paper for Right centralizers are also true for left centralizers because of left-right symmetry. Definition 1.1 : A semiring is a non empty set 푆 followed with two binary operations addition and multiplication such that (푆, +) is a semigroup (푆, . ) is a semigroup Multiplication distributes over addition from either side. Definition 1.2: The semiring (푆, +, . ) is said to be semiring with additive identity 0 such that 푥 + 0 = 푥 = 0 + 푥 for all 푥 휖 푆 and addition is commutative. A semiring (푆, +, . ) is said to have an identity element 1, if 1 ≠ 0 such that 1. 푥 = 푥 = 푥 .1 for all 푥 휖 푆 and multiplication is commutative. Semiring (푆, +, . ) is said to be commutative if both (푆 +) and (푆, . ) are commutative. Definition 1.3 : A Semiring 푆 is said to be prime if 푥푆푦 = 0 implies 푥 = 0 or 푦 = 0 for all 푥, 푦 휖 푆 A Semiring 푆 is said to be semiprime if 푥푆푥 = 0 implies 푥 = 0 for all 푥 휖 푆. A semiring 푆 is said to be 2-torsion free if 2 푥 = 0 implies 푥 = 0 for all 푥 휖 푆. On Right Centralizers of Semiprime Semirings 21 Definition 1.4: A semiring 푆 is said to be commutative semiring if 푥 푦 = 푦푥 for all 푥 , 푦 휖 푆. Then the set 푍(푠) = {푥 휖 푆, 푥푦 = 푦푥 for all 푦 휖 푆} is called the centre of the semiring 푆. Definition1.5: For any fixed 푎 휖 푆 the mapping 푇(푥) = 푎푥 is a left centralizer and 푇(푥) = 푥푎 is a right centralizer. An additive mapping 푇: 푆 → 푆 is a left (Right) centralizer if 푇(푥푦) = 푇(푥)푦, ((푇(푥푦) = 푥푇(푦)) for all 푥 , 푦 휖 푆. Definition 1.6: 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] for all 푥 , 푦, 푧 휖 푆. According to [13] 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 Definition 1.7: A mapping h: S→S is called centralizing (skew centralizing) if [ℎ(푥), 푥] 휖 푍(푆). (ℎ(푥)푥 + 푥ℎ(푥) 휖 푍( 푆)) for all 푥휖 푆 . A mapping ℎ of 푎 semiring 푆 into itself is said to be commuting if [ℎ(푥), 푥] = 0 and skew commuting if (ℎ(푥)푥 + 푥ℎ(푥) = 0). Obviously every commuting( skew commuting) mapping ℎ: 푆 → 푆 is centralizing (skew centralizing). We recall if ℎ: 푆 → 푆 is commuting then [ℎ(푥), 푦] = [푥, ℎ(푦)] for all 푥, 푦휖 푆. A mapping ℎ: 푆 → 푆 is called central if ℎ(푥)휖 푍(푆)for all 푥휖 푆. Definition 1.8: Let 퐴 be a subset of 푆. If 푥푦 + 푦푥 휖퐴 for all 푥, 푦 휖 퐴 then 퐴 is called a Jordan subring of 푆. Notation 1.1: We consider semiring with absorbing Zero and with identity 1 ≠ 0. 22 D.Mary Florence, R. Murugesan and P. Namasivayam 2 MAIN RESULTS Theorem 2.1: Let 푆 be a 2- torsion free semiprime semiring and 퐴 be a Jordan subring of 푆. If an additive mapping 퐻 of 푆 into itself is centralizing on 퐴, then 퐻 is commuting on 퐴. Proof: By the assumption [퐻(푥), 푥] 휖 푍(푆). The linearization gives [퐻(푥), 푦] + [퐻(푦), 푥] 휖푍 for all 푥, 푦 휖 퐴. In particular if 푦 = 푥2 one obtains that 2[퐻(푥), 푥]푥 + [퐻(푥2), 푥] 휖푍 for all 푥 휖퐴. (1) On the other hand we have [퐻(푥2), 푥2] 휖푍 for all x ϵA That is [퐻(푥2), 푥]푥 + 푥 [퐻(푥2), 푥] 휖 푍 (2) Let 푥 휖퐴 be a fixed element and let 푧 = [퐻(푥), 푥] 휖푍 , 푎 = [퐻(푥2), 푥] in (1) we obtain [퐻(푥), 2푧푥 + 푎] = 0. We are forced to show that 푧 = 0. [퐻(푥), 2푧푥] + [퐻(푥), 푎] = 0 2푧[퐻(푥), 푥] + 2[퐻(푥), 푧]푥 + [퐻(푥), 푎] = 0 [퐻(푥), 푎] = 2푧′2 (3) Now using (2), one can write [퐻(푥), 푎푥 + 푥푎] = 0 and apply (3) we get [퐻(푥), 푎]푥 + 푎[퐻(푥), 푥] + [퐻(푥), 푥]푎 + 푥[퐻(푥), 푎] = 0 2za + 4 z′2x = 0 za = 2z2x multiplying (3) by z we obtain z[H(x), a] = 2푧푧′2 [H(x), za] + [H(x), z]a′ = 2푧푧′2 Substituting za = 2z2x in the above [H(x), 2z2x] = 2푧푧′2 2{[H(x), z2]x + z2[H(x), x]} = 2푧푧′2 Adding 2z3 on both sides, we get 2z3 + 2z3 = 0 2z3 = 0. 푆 is 2 torsion free semiprime semiring z3 = 0, Since the centre of a semiprime semiring contains no non zero nilpotent elements. This leads us to conclude that 푧 = 0. This proves the theorem. Theorem 2.2: Let 푆 be a semiprime semiring. If 푇: 푆 → 푆 be a centralizing Right Centralizer. Then 푇 is commuting. Proof: If S is 2 torsion free, then by taking 퐴 = 푆 in theorem (2.1) it follows immediately that 푇 is commuting. If 푆 is not a 2 torsion free semiprime semiring, then On Right Centralizers of Semiprime Semirings 23 2[푥, 푇(푥)] = 0 for all x ϵS (4) Substituting 푥 + 푦 for 푥 , we arrive at 2[푥, 푇(푦)] + 2[푦, 푇(푥)] = 0 [푥, 푇(푦)] = [푦, 푇(푥)]′ for all 푥, 푦 휖푆. (5) linearising the assumption [푥, 푇(푥)] 휖 푍(푠)] and using (5) we obtain [푥, 푇(푦)] + [푦, 푇(푥)] 휖 푍(푠) (6) [([푥, 푇(푦) ] + [푦, 푇(푥) ] ), 푥] = 0 푓표푟 푎푙푙 푥, 푦 휖푆 [푥, 푇(푦)]푥 + [푦, 푇(푥)] + 푥′[푥, 푇(푦)] + 푥′[푦, 푇(푥) ] = 0 [푥, 푇(푦)]푥 + [푦, 푇(푥) ]푥 + 푥[푦, 푇(푥) ] + 푥[푥, 푇(푦) ] = 0 (7) From assumption we found the result [푥, 푇(푥)]푦 = 푦[푥, 푇(푥)] Applying the above result in relation (4) and is reached at [푥, 푇(푥)]푦 + 푦[푥, 푇(푥) ] = 0 (8) Adding expressions (7) and (8) then arranged, we get [푥푦 + 푦푥, 푇(푥)] + [푥2, 푇(푦) ] = 0 (9) Now using 푥푦 for y in the above relation we arrive at [푥푥푦 + 푥푦푥, 푇(푥)] + [푥2, 푇(푥푦) ] = 0 [푥, 푇(푥) ](푥푦 + 푦푥) + 푥[푥푦 + 푦푥, 푇(푥) ] + 푥[푥2, 푇(푦) ] = 0 (10) Finally combining the last relation with (9), we get [푥, 푇(푥) ](푥푦 + 푦푥) = 0 for all 푥, 푦 휖푆 [푥, 푇(푥) ](푥푦 + 푦′푥 + 푦푥 + 푦푥) = 0 [푥, 푇(푥) ](푥푦 + 푦′푥) + 2[푥, 푇(푥) ]푦푥 = 0 [푥, 푇(푥) ](푥푦 + 푦′푥)] = 0.