Riv. Mat. Univ. Parma, Vol. 10, No. 1, 2019

On derivations of rings and Banach algebras involving anti-commutator

Pages: 85-97
Accepted in revised form: 18 March 2019
Mathematics Subject Classification (2010): 46J10, 16N20, 16N60, 16W25.
Keywords: Semiprime ring, derivation, maximal right ring of quotient, Banach algebra.
[a]: Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India

Abstract: In the present paper we obtain commutativity of a semiprime ring $$\ \mathcal{R} \$$ which admits a derivation $$d$$ such that either $$\ (d(x^m \circ y^n))^\ell \pm (x^p \circ_k y^q) = 0 \$$ for all $$\ x,y\in \mathcal{R} \$$ or $$\ (d(x^m) \circ d(y^n))^\ell \pm (x^p \circ_k y^q) = 0 \$$ for all $$\ x,y\in \mathcal{R} \$$, where $$\ m,n,p,q,k,\ell \$$ are fixed positive integers. Finally, we apply the above purely ring theoretic results to Banach algebras and obtain a noncommutative version of the Singer-Wermer theorem. In particular, we prove that if $$\ \mathfrak{B} \$$ is a noncommutative Banach algebra which admits a continuous linear derivation $$d:\mathfrak{B}\rightarrow \mathfrak{B}$$ such that either $$\ (d(x^m \circ y^n))^\ell \pm (x^p \circ_k y^q)\in rad(\mathfrak{B}) \$$ or $$\ (d(x^m) \circ d(y^n))^\ell \pm (x^p \circ_k y^q)\in rad(\mathfrak{B}) \$$ for all $$\ x,y\in \mathfrak{B} \$$, where $$\ m,n,p,q,k,\ell \$$ are fixed positive integers, then $$\ d(\mathfrak{B})\subseteq rad(\mathfrak{B}) \$$.

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