abstract
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In this thesis we make an extensive study of the algebraic solutions of the functional equation
[equation removed]
where the unknown function [equation removed] maps a ring to an abelian group G.
After proving some general results about the solutions of the equation, we study it over rings generated by their units, over number rings, and over polynomial rings. We find that over a large class of rings, the equation is equivalent to Cauchy's functional equation, and we give ideal-theoretic criteria to specify when it is not.
Our methods involve a wide variety of techniques and results from algebra and algebraic number
We complete our study with an a class of functional equations which generalizes the above equation.