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In this paper, we develop the theory of modules over (A, δ), where A is an algebra and δ : A −→ A is a derivation. Our approach is heavily influenced by Lie derivative operators in noncommutative geometry, which make the Hochschild homologies H H • (A) of A into a module over (A, δ). We also consider modules over (A,), where = { n } n≥0 is a higher derivation on A. Further, we obtain a Cartan homotopy formula for an arbitrary higher derivation on A.

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