If R is a Euclidean domain, then there are universal side divisors in R

How to prove that a Euclidean Domain has universal side divisors

The best way to prove that is to use the definition of a Euclidean Domain straightly and by using the definition of universal side divisors.

Prove that if R is a Euclidean domain and not a field, then there are universal side divisors in R

Proof: let R be a Euclidean domain and let b \in R - \widetilde{R} = R - R^{\times} \cup \{0\} be a universal side divisor of minimal norm. This means that we took a universal side divisor which we are sure that R has no smaller norms of universal side divisors than the norm of b. By the definition of the Euclidean Domain, we can say
x = qb + r \quad \text{where} \quad r = 0 \text{ or } N(r) < N(b).
Now, we need to show that r \in R^{\times} \cup \{0\}. It is obviously true that if r = 0, then r \in R^{\times} \cup \{0\}. If N(r) < N(b), then we do know that r can't be a universal divisor since b is the smallest norm that is a universal divisor, which implicates that r \not \in R - \widetilde{R}. So r must be a unit and therefore r \in R^{\times} \cup \{0\}, which proves this proof.


It is crucial to notice that we are curious if there are universal side divisors, so showing one is perfectly fine.

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