Note di Matematica ISSN 1123-2536, e-ISSN 1590-0932 Note Mat. 40 (2020) no. 1, 73–79. doi:10.1285/i15900932v40n1p73 On rings and Banach algebras with skew derivations Nadeem ur Rehman Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India nu.rehman.mm@amu.ac.in Received: 21.7.2019; accepted: 21.10.2019. Abstract. In the present paper, we investigate the commutativity of a prime Banach algebra with skew derivations and prove that if A is prime Banach algebra and A has a nonzero continuous linear skew derivation F from A to A such that [F(x m ), F(y n )] - [x m , y n ] ∈Z(A) for an integers m = m(x, y) > 1 and n = n(x, y) > 1 and sufficiently many x, y, then A is commutative. Keywords: Prime Banach algebra, skew derivation. MSC 2000 classification: primary 16W25, secondary 46J10 Introduction Several theorems in ring theory, mostly due to Herstein, are devoted to showing that certain rings must be commutative as a consequence of conditions which are seemingly too weak to imply commutativity. Consider the following theorem of Herstein [9, p 412] which states that a ring R is commutative if for each x and y ∈ R there is a positive integer n(x, y) > 1 such that x n(x,y) − x permutes with y. This research is motivated by the work of Ali and Khan [1] and Yood [16]. Throughout this manuscript A represents a Banach algebra over the complex field, Z (A) denote the center of A and M be a closed linear subspace of A. Recall that an algebra A is said to be prime if for any a, b ∈A, aAb = (0) implies that a = 0 or b = 0, and A is semiprime if for any a ∈A, aAa = (0) implies a = 0. We shall use several times the readily fact. Let p(t)= ∑ n r=0 b r t r be a polynomial in the real variable t with coefficients in A. If p(t) ∈ M for all t in an infinite subset of the reals, then every b r lies in M . A linear map F of A into itself is called a linear derivation if F(xy)= F(x)y + xF(y) for all x, y ∈A. Let σ be an automorphism of A. A linear map F : A→A is called a linear skew-derivation if F(xy)= F(x)y + σ(x)F(y) for all x, y ∈A. When σ = I A on A, linear skew-derivation is simply an ordinary linear derivation. For σ = I A , http://siba-ese.unisalento.it/ 2020 Universit`a del Salento brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by ESE - Salento University Publishing