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A type of generalized higher derivation consisting of a collection of self-mappings of a ring associated with a monoid, and here called a D-structure, is studied. Such structures were previously used to define various kinds of 'skew' or 'twisted' monoid rings. We show how certain gradings by monoids define D-structures. The monoid ring defined by such a structure corresponding to a group-grading is the variant of the group ring introduced by Nȃstȃsescu, while in the case of a cyclic group ofdoi:10.1017/s000497271100308x fatcat:ihw3b4u2yfbcrh7avmnbpbufue