On derivations involving prime ideals and commutativity in rings

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.

Some results on derivations and generalized derivations in rings

“…Since R is prime and b ̸ ∈ Z(R), the last equation reduces to h(z) = 0. Now suppose that h(z) = 0 for all z ∈ Z(R), then (11) becomes…”

“…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. 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 [1] and [11]). A well known result of Posner [12] 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.…”

On Multiplicative (Generalized)-Derivation Involving Semiprime Ideals

“…Numerous studies have revealed that the global structure of A is typically related to the behavior of additive mappings formed on A. Recent work has investigated the commutativity of the factor ring A/P, where P is the prime ideal of any arbitrary ring A, using algebraic identities in P, including derivations and generalized derivations (see [6][7][8][9][10][11]).…”

Some commutativity criteria involving endomorphism conditions on prime ideals

“…Moreover, many of obtained results extend other ones proven previously just for the action of the considered mapping on the entire ring. In this direction, the recent literature contains numerous results on commutativity in prime and semi-prime rings admitting suitably constrained derivations and generalized derivations (for example, see [7][8][9]12]). The purpose of this paper is to study the commutativity of a quotient ring R/P where R is an arbitrary ring and P is a prime ideal of R. The originality in this work is that we will consider endomorphisms of R satisfying some specific algebraic identities involving P and without primeness (semi-primeness) assumption on the ring R.…”