Euclidean Rings

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• Ring Theory:Euclidean Ring: Prove that a necessary and sufficient condition that the element 'a' in the Euclidean ring is a unit is that d(a) = d(1). Or, Prove that the element 'a' in the Euclidean ring is a unit if and only if d(a) = d(1).

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  It seems to me that the main difference between the two rings is in the norm and, therefore, in their units. I have updated my file, but I can't find any mistake in my proof.