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On the centroid of prime semirings

We define and study the extended centroid of a prime semiring. We show that the extended centroid is a semifield and give some properties of the centroid of a right multiplicatively cancellable prime semiring. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 577 – 584 ¨ ITAK ˙ c TUB doi:10.3906/mat-1105-8 On the centroid of prime semirings 2 ¨ ¨ Hasret YAZARLI,1, ∗ Mehmet Ali OZT URK Cumhuriyet University, Faculty of Arts and Sciences, Department of Mathematics, 58140 Sivas, Turkey Adıyaman University, Faculty of Arts and Sciences, Department of Mathematics, Adıyaman, Turkey Received: 09.05.2011 • Accepted: 13.06.2012 • Published Online: 12.06.2013 • Printed: 08.07.2013 Abstract: We define and study the extended centroid of a prime semiring. We show that the extended centroid is a semifield and give some properties of the centroid of a right multiplicatively cancellable prime semiring. Key words: Semiring, prime ideal, prime semiring, quotient semiring 1. Introduction Semirings abound in the mathematical world around us. Indeed, the first mathematical structure we encounter— the set of natural numbers—is a semiring. Historically, semirings first appear implicitly in [3] and later in [8], [6], [10] and [7], in connection with the study of ideals of a ring. They also appear in [4] and [5] in connection with the axiomatization of the natural numbers and nonnegative rational numbers. Over the years, semirings have been studied by various researchers either in their own right, in an attempt to broaden techniques coming from semigroup theory or ring theory, or in connection with applications. In [9] Martindale first constructed ¨ urk and Jun introduced the extended centroid of a for any prime ring R a “ring of quotients” Q . After, Ozt¨ prime Γ-ring and obtained some results in Γ-ring M with derivation which was related to Q , and the quotient Γ-ring of M [11, 12]. In this paper, we define and study the extended centroid of a prime semiring. 2. Preliminaries A semiring is a nonempty set R on which operations of addition and multiplication have been defined such that the following conditions are satisfied: (1) (R,