# 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

\begin{align*} x = qb + r \quad \text{where} \quad r = 0 \text{ or } N(r) < N(b). \end{align*}

## Conclusion

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