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We review the notion of internal hom for a monoidal category. The discussion includes the endomorphism monoid, the convolution monoid, the internal hom for functor categories (which includes the category of modules over a monoid algebra). We also discuss the enriched counterpart of the tensor-hom adjunction, which gives rise to the notion of power and copower.

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 Năstăsescu, while in the case of a cyclic group of order two, the form of the D-structure itself yields some gradability criteria of Bakhturin and Parmenter. A partial description is obtained of the D-structures associated with infinite cyclic monoids.

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Appendix B - Internal hom for monoidal categories

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GENERALIZED HIGHER DERIVATIONS

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Appendix B - Internal hom for monoidal categories

Summary

GENERALIZED HIGHER DERIVATIONS