In an earlier paper we discussed the composition of derivations of order 1 on a commutative ring R, showing that (i) the composition of n derivations of order 1 yields a derivation of order at most n, and (ii) under additional conditions on R the composition of n derivations of order exactly 1 forms a derivation of order exactly n. In the present paper we consider the composition of derivations of any orders on rings. We show that on any commutative ring R the composition of a derivation of order at most n with a derivation of order at most m results in a derivation of order at most \(n+m\). If R is an integral domain of sufficiently large characteristic, then the composition of a derivation of order exactly n with a derivation of order exactly m results in a derivation of order exactly \(n+m\). As in the previous paper, the results are proved using Leibniz difference operators.
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Ebanks, B.: Characterizing ring derivations of all orders via functional equations: results and open problems. Aequ. Math. 89, 685–718 (2015)
Ebanks, B.: Derivations and Leibniz differences on rings. Aequ. Math. (2018) https://doi.org/10.1007/s00010-018-0601-4
Ebanks, B.: Functional equations characterizing derivations: a synthesis. Results Math 73(3), 13 (2018). https://doi.org/10.1007/s00025-018-0881-y. Art. 120
Gselmann, E., Kiss, G., Vincze, C.: On functional equations characterizing derivations: methods and examples. Results Math. 73(2), 27 (2018). https://doi.org/10.1007/s00025-018-0833-6. Art. 74
Kiss, G., Laczkovich, M.: Derivations and differential operators on rings and fields. Ann. Univ. Sci. Budapest. Sect. Comput. 48, 31–43 (2018)
I am very grateful to the reviewer for several comments and suggestions which have clarified and improved the presentation of these results.
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Ebanks, B. Derivations and Leibniz differences on rings: II. Aequat. Math. 93, 1127–1138 (2019). https://doi.org/10.1007/s00010-018-0630-z
Mathematics Subject Classification
- Commutative ring
- Integral domain
- Derivations of higher order
- Leibniz difference operator