Riv. Mat. Univ. Parma, Vol. 10, No. 1, 2019
Mohammad Ashraf ^{[a]} and Bilal Ahmad Wani ^{[a]}
On derivations of rings and Banach algebras involving anticommutator
Pages: 8597
Received: 3 December 2018
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.
Authors address:
[a]: Department of Mathematics, Aligarh Muslim University, Aligarh202002, India
Full Text (PDF)
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 SingerWermer 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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