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.

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The purpose of this paper is to study derivations and generalized derivations in prime rings satisfying certain differential identities. Some well-known results characterizing commutativity of prime rings have been generalized. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

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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.