On the identity \(D^{3}+D^{2}=G\) in semiprime rings

Benjamin Marcen; Joso Vukman

doi:10.56754/0719-0646.2901.001

Authors

Benjamin Marcen Faculty of Logistics, University of Maribor, Mariborska cesta 7, 3000 Celje, Slovenia https://orcid.org/0000-0001-8436-6672
Joso Vukman Faculty of Natural sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia https://orcid.org/0009-0006-8596-7246

MSC:

16N60, 16W25

DOI:

https://doi.org/10.56754/0719-0646.2901.001

Keywords:

Prime ring, semiprime ring, Banach algebra, derivation

Abstract

The purpose of this paper is to prove the following result. Let \(R\) be a \(2-\)torsion free semiprime ring and let \(D, G : R \rightarrow R\) be derivations. Suppose that \(D^{3}(x)+D^{2}(x)=G(x)\) holds for all \(x \in R\). In this case \(D=0\). The result is related to the classical theory of derivations initiated by Posner.

References

[1] A. Giambruno and I. N. Herstein, “Derivations with nilpotent values,” Rend. Circ. Mat. Palermo, vol. 30, pp. 199–206, 1981, doi: 10.1007/BF02844306.

[2] I. N. Herstein, Rings with Involution, ser. Chicago Lectures in Mathematics. Chicago and London: University of Chicago Press, 1976.

[3] E. C. Posner, “Derivations in prime rings,” Proc. Am. Math. Soc., vol. 8, pp. 1093–1100, 1958, doi: 10.2307/2032686.

[4] J. Vukman, “A result concerning derivations in Banach algebras,” Proc. Am. Math. Soc., vol. 116, no. 4, pp. 971–975, 1992, doi: 10.2307/2159476.