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On b-generalized derivations and commutativity of prime rings
Derivadas b-generalizadas y conmutatividad de anillos primos
Palabras clave:
Prime ring, Martindale quotient ring, b-Generalized derivation (en)Anillos primo, anillo de cocientes de Martindale, derivadas b-generalizadas (es)
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Let A be a prime ring, Z(A) its center, Q its right Martindale quotient ring, C its extended centroid, ψ a non-zero b-generalized derivation of A with associated map ξ. In this article, we prove that: (i) If [ψ(x), ψ(y)] = 0 for all x, y ∈ A, then A is either commutative or there exists q ∈ Q such that ξ = ad(q), ψ(x) = -bxq, and qb = 0. (ii) If ψ(x) ◦ ψ(y) = 0 for all x, y ∈ A, then A is either commutative with char(A) = 2 or there exists q ∈ Q such that ψ(x) = -bxq and qb = 0. Additional results are established for cases involving [ξ(x), ψ(x)] = 0 or ξ(x)◦ψ(x) = 0, where char(A) = 2. Furthermore, we give some examples that show the importance of the hypotheses of our theorems.
Sea A un anillo primo, Z(A) su centro, Q su anillo de cocientes de Martindale por derecha, C su centroide extendido, ψ una derivada b-generalizada de A con mapa asociado ξ. En este artículo probamos los siguientes resultados: (i) Si [ψ(x), ψ(y)] = 0 para todo x, y ∈ A, entonces o A es conmutativo o existe q ∈ Q tal que ξ = ad(q), ψ(x) = -bxq, y qb = 0. (ii) Si ψ(x) ◦ ψ(y) = 0 para todo x, y ∈ A, entonces o A es conmutativo con char(A) = 2 o existe q ∈ Q tal que ψ(x) = -bxq y qb = 0. También se analizan los casos donde [ξ(x), ψ(x)] = 0 o ξ(x) ◦ ψ(x) = 0, donde char(A) = 2. Se incluyen ejemplos que ilustran la importancia de las hipótesis de los teoremas.
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