# Derivations and Leibniz differences on rings: II

• Bruce Ebanks
Article

## 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.

## Keywords

Commutative ring Integral domain Derivation Derivations of higher order Leibniz difference operator

## Mathematics Subject Classification

39B52 13N15 39B72

## Notes

### Acknowledgements

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

## References

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