The new concept"derimorphism"generalizing both derivation and homomor-phism is defined.When a derimorphism is invertible,its inverse is a Rota-Baxter operator.The general theory of derimorphism is established.The classification of all derimorphisms over an associative unital algebra is obtained.Contrary to the nonexistence of nontriv-ial positive derivations,it is shown that nontrivial positive derimorphisms do exist over any pair of opposite orderings over R[x],the lattice-ordered full matrix algebra and upper triangular matrix algebra over a totally ordered field.
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