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Derivations, generalized derivations and derivations of period 2 in rings

The aim of this article is to discuss the existence of certain kinds of derivations and -derivations that are of period 2. Moreover, we obtain the form of generalized reverse derivations and generalized left derivations of period 2. | Turk J Math (2018) 42: 2664 – 2671 © TÜBİTAK doi:10.3906/mat-1805-111 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Derivations, generalized derivations, and *-derivations of period 2 in rings Hesham NABIEL∗, Department of Mathematics, Faculty of Science Al-Azhar University, Nasr City, Cairo, Egypt Received: 22.05.2018 • Accepted/Published Online: 06.08.2018 • Final Version: 27.09.2018 Abstract: The aim of this article is to discuss the existence of certain kinds of derivations and *-derivations that are of period 2. Moreover, we obtain the form of generalized reverse derivations and generalized left derivations of period 2 . Key words: Maps of period 2, derivations, generalized derivations, *-derivations, prime rings, semiprime rings 1. Introduction Throughout this paper, R will represent an associative ring with center Z(R) . An ideal U of R is said to be central ideal if U ⊆ Z(R) . Given an integer n ≥ 2, a ring R is said to be n -torsion free if for x ∈ R , nx = 0 implies x = 0 . For x, y ∈ R , the symbol [x, y] stands for the commutator xy − yx . R is said to be domain if for a, b ∈ R , ab = 0 implies a = 0 or b = 0 . A domain with identity is called a unital domain. R is said to be prime if for a, b ∈ R , aRb = {0} implies a = 0 or b = 0 , and is said to be semiprime if for a ∈ R , aRa = {0} implies a = 0 . Its clear that every domain is prime. An additive mapping d : R → R is called a derivation (Jordan derivation, respectively) if d(xy) = d(x)y + xd(y) for all x, y ∈ R ( d(x2 ) = d(x)x + xd(x) for all x ∈ R , respectively). As in [9] by Bell and Daif and in [14] by Gölbaşi and Kaya, a right (left, respectively) generalized derivation F of R is an additive map of R associated with a derivation d of R such that F (xy) = F (x)y + xd(y) for all x, y ∈ R ( F (xy) = xF (y) + d(x)y for all x, y ∈ R , respectively). If F is both a right and left generalized derivation with the same associated derivation, then F is .