Abstract
Let R be a commutative ring and n a positive integer. We show that the composition of n derivations of order 1 results in a derivation of order n on R. If in addition R is an integral domain of characteristic 0, then the composition of n nontrivial derivations of order 1 forms a nontrivial derivation of order n. This is also true for integral domains of characteristic larger than n!, but not for integral domains of characteristic n!, nor for commutative rings (even of characteristic 0) which are not integral domains. We prove our results by the use of Leibniz differences. One corollary is that nontrivial derivations of all orders exist on \({\mathbb R}\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ebanks, B.: Characterizing ring derivations of all orders via functional equations: results and open problems. Aequ. Math. 89, 685–718 (2015)
Kiss, G., Laczkovich, M.: Derivations and differential operators on rings and fields. Ann. Univ. Sci. Bp. Sect. Comput. arXiv:1803.01025
Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality, 2nd edn. Birkhäuser, Basel (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ebanks, B. Derivations and Leibniz differences on rings. Aequat. Math. 93, 629–640 (2019). https://doi.org/10.1007/s00010-018-0601-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-018-0601-4