On derivations involving prime ideals and commutativity in rings
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On derivations involving prime ideals and commutativity in rings A. Mamouni1 · L. Oukhtite2 · M. Zerra2 Accepted: 14 September 2020 © Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract The principal aim of this paper is to study the structure of quotient rings R/P where R is an arbitrary ring and P is a prime ideal of R. Especially, we will establish a relationship between the structure of this class of rings and the behaviour of derivations satisfying algebraic identities involving prime ideals. Some well-known results characterizing commutativity of (semi)-prime rings have been generalized. Keywords Prime ideal · Commutativity · Derivation Mathematics Subject Classification 16N60 · 16W10 · 16W25
1 Introduction Throughout this paper R will represent an associative ring with center Z(R). Recall that a proper ideal P of R is said to be prime if for any x, y ∈ R , xRy ⊆ P implies that x ∈ P or y ∈ P. The ring R is a prime ring if and only if (0) is a prime ideal of R. For any x, y ∈ R the symbol [x, y] will denote the commutator xy − yx ; while the symbol x◦y will stand for the anti-commutator xy + yx . An additive mapping d ∶ R ⟶ R is a derivation if d(xy) = d(x)y + xd(y) for all x, y ∈ R . Let a ∈ R be a fixed element. Communicated by Sergio R. López-Permouth. * A. Mamouni [email protected] L. Oukhtite [email protected] M. Zerra [email protected] 1
Department of Mathematics, Faculty of Science and Technology, University Moulay Ismaïl, Box 509‑Boutalamine, Errachidia, Morocco
2
Department of Mathematics, Faculty of Science and Technology, University Sidi Mohamed Ben Abdellah, Fez, Morocco
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
A map d ∶ R ⟶ R defined by d(x) = [a, x] = ax − xa , x ∈ R , is a derivation on R, which is called inner derivation defined by a. Recently, many results in literature indicate how the global structure of a ring R is often tightly connected to the behaviour of additive mappings defined on R (for example, see [3, 7, 18] and [11]). Herstein [10] showed that a 2-torsion free prime ring R must be a commutative integral domain if it admits a nonzero derivation d satisfying [d(x), d(y)] = 0 for all x, y ∈ R , and if the characteristic of R equals two, the ring R must be commutative or an order in a simple algebra which is 4-dimensional over its center. Several authors have proved commutativity theorems for prime rings admitting derivations which are centralizing on R. We begin recalling that a mapping f ∶ R ⟶ R is called centralizing on R if [f (x), x] ∈ Z(R) for all x ∈ R . A well known result of Posner [17] states that if d is a derivation of the prime ring R such that [d(x), x] ∈ Z(R) , for any x ∈ R , then either d = 0 or R is commutative. In [13] Lanski generalizes the result of Posner to a Lie ideal. More recently several authors considered similar situation in the case the derivation d is replaced by a generalized derivation. More specifically an additive map F ∶ R ⟶ R is said to be a generalized derivati
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