On Centrally Extended Reverse and Generalized Reverse Derivations
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DOI: 10.1007/s13226-020-0456-y
ON CENTRALLY EXTENDED REVERSE AND GENERALIZED REVERSE DERIVATIONS Susan F. El-Deken∗ and Mahmoud M. El-Soufi∗∗ ∗ Department
of Mathematics, Faculty of Science, Helwan University, Ain Helwan, 11790, Helwan, Egypt
∗∗ Department
of Mathematics, Faculty of Science, Fayoum University, Egypt and
Department of Mathematics, Faculty of Science, Al Baha University, Kingdom of Saudi Arabia e-mails: [email protected]; [email protected] (Received 15 May 2018; after final revision 22 February 2019; accepted 20 June 2019) In this paper, we define the notions of centrally-extended reverse derivations and centrally-extended generalized reverse derivations and discuss the relationship between these mappings and the reverse derivations as well as the generalized reverse derivations. Also, we give some commutativity results. Key words : Reverse derivations; generalized reverse derivations; centrally-extended reverse derivations; centrally-extended generalized reverse derivations. 2010 Mathematics Subject Classification : 16N60, 16W25, 16U80.
1. I NTRODUCTION In 2016, Bell and Daif [2] have introduced the concept of centrally-extended derivations of a ring R. They investigated the existence of centrally-extended derivations and studied their effect on the center Z(R). A map D : R → R, satisfying D(x + y) − D(x) − D(y) ∈ Z(R) and D(xy) − D(x)y − xD(y) ∈ Z(R) for all x, y ∈ R, is called a centrally-extended derivation (CE-derivation). We introduce the concepts of centrally-extended reverse derivations (CE-reverse derivations) and centrally-extended generalized reverse derivations (CE-generalized reverse derivations). A CEreverse derivation is a mapping D : R → R, satisfying D(x + y) − D(x) − D(y) ∈ Z(R) and
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SUSAN F. EL-DEKEN AND MAHMOUD M. EL-SOUFI
D(xy) − D(y)x − yD(x) ∈ Z(R) for all x, y ∈ R. A CE-l-generalized reverse derivation (CE-rgeneralized reverse derivation) is a mapping F : R → R, satisfying F (x+y)−F (x)−F (y) ∈ Z(R) and F (xy)−F (y)x−yD(x) ∈ Z(R), (F (xy)−D(y)x−yF (x) ∈ Z(R)) for all x, y ∈ R, where D is a CE−reverse derivation of R. A CE-generalized reverse derivation is both a CE-l-generalized reverse derivation and a CE-r-generalized reverse derivation. Throughout this paper R will denote an associative ring with center Z(R). A ring R is said to be prime if xRy = {0} implies either x = 0 or y = 0. A ring R is called semiprime if xRx = {0} implies x = 0. A(R) = {x ∈ R : xR = 0} denotes the two-sided annihilator of R. The symbol [x, y] will denote the commutator xy − yx. Basic commutator identities [xy, z] = x[y, z] + [x, z]y and [x, yz] = [x, y]z + y[x, z] shall be used extensively. Let M be a subset of a ring R. A mapping f : R → R is said to preserve M if f (M ) ⊆ M . The set M R is said to be the right multiplication of M by R. We say that f : R → R preserves the right multiplication of M by R if f (M R) ⊆ M . Note that if R is unital, then f preserves M . Let N be the set of nilpotent elements of R, and call R reduced if N = {0}. An additive map D : R → R is called a der
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