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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Acknowledgements

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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Correspondence to Bruce Ebanks.

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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

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Mathematics Subject Classification

  • 39B52
  • 13N15
  • 39B72

Keywords

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