# Derivations and Leibniz differences on rings: II

## Abstract

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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## References

1. 1.

Ebanks, B.: Characterizing ring derivations of all orders via functional equations: results and open problems. Aequ. Math. 89, 685–718 (2015)

2. 2.

Ebanks, B.: Derivations and Leibniz differences on rings. Aequ. Math. (2018) https://doi.org/10.1007/s00010-018-0601-4

3. 3.

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

4. 4.

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

5. 5.

Kiss, G., Laczkovich, M.: Derivations and differential operators on rings and fields. Ann. Univ. Sci. Budapest. Sect. Comput. 48, 31–43 (2018)

## Acknowledgements

I am very grateful to the reviewer for several comments and suggestions which have clarified and improved the presentation of these results.

## Author information

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### Corresponding author

Correspondence to Bruce Ebanks.