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Prime ideals and generalized derivations with central values on rings Mamouni Abdellah1 · Oukhtite Lahcen2 · Zerra Mohammed2  Received: 12 July 2020 / Accepted: 5 November 2020 © The Author(s) 2020

Abstract Our goal in the present paper is to study a connection between the commutativity of rings and the behaviour of its generalized derivations. More specifically, we investigate commutativity of quotient rings R/P where R is any ring and P is a prime ideal of R which admits generalized derivations satisfying certain algebraic identities acting on prime ideal P without the primeness (semi-primeness) assumption on the considered ring. This approach allows us to generalize some well known results characterizing commutativity of rings. Keywords  Prime ideal · Commutativity · Generalized derivations Mathematics Subject Classification  16N60 · 16U80 · 16W25

1 Introduction Throughout this paper, R denotes an associative ring with center Z(R). Recall that an ideal P of R is said to be prime if P ≠ R and for x, y ∈ R , xRy ⊆ P implies that x ∈ P or y ∈ P . A prime ideal P of R minimal if P does not properly include any prime ideals of R. The ring R is a prime ring if and only if (0) is the only minimal prime ideal of R. The Lie product of two elements x and y of R is [x, y] = xy − yx ; while the symbol x◦y will stand for the anticommutator xy + yx. A map d ∶ R → R is a derivation of a ring R if d is additive and satisfies the Leibnitz rule; d(xy) = d(x)y + xd(y) for all x, y ∈ R. Over the last years, several authors have investigated the relationship between the commutativity, the structure of the ring R and certain special types of maps on R (see for example; [4, 8, 13]). Herstein [7] showed that a prime ring R with nonzero derivation d * Zerra Mohammed [email protected] Mamouni Abdellah [email protected] Oukhtite Lahcen [email protected] 1

Department of Mathematics, Faculty of Sciences, University Moulay Ismaïl, Meknes, Morocco

2

Department of Mathematics, Faculty of Science and Technology of Fez, University S. M. Ben Abdellah, Box 2202, Fez, Morocco



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satisfying d(x)d(y) = d(y)d(x) for all x, y ∈ R , must be a commutative integral domain if its characteristic is not two, and, if the characteristic equals two, 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 first recall that a mapping f ∶ R → R is called centralizing (on R) if [f (x), x] ∈ Z(R) for all x ∈ R ; in the special case where [f (x), x] = 0 for all x ∈ R , the mapping f is said to be commuting on R. In [13], Posner proved that if a prime ring R admits a nonzero derivation d such that [d(x), x] ∈ Z(R) for all x ∈ R , then R is commutative. Since then many authors have extended the Posner’s result in several directions. A considerable number of researchers have investigated the ideals (Jordan ideal, Lie ideal...) in rings as well as t