# How to show that every ideal in a Euclidean Domain is principal

We need to show any ideal I of the Euclidean Domain R is principal, i.e., I = (x) where x is any nonzero element of I of minimum norm.## Prove that every ideal in a Euclidean Domain is principal

Let R be an Euclidean Domain and I its arbitrary ideal. It is obvious that when I is zero ideal, then I = (0). Now assume I is a non-empty ideal and let b \neq 0 \in I be its minimal norm. To show that b exists, we need to show that the following set has a minimal element:\begin{align*} \{N(y) \ | \ y \in I\}. \end{align*}

“I \subseteq (b)“: Since the assumption was that R is an Euclidean Domain, R also possess the Division Algorithm. So take the element a \in I such that a = qb + r with r = 0 or N(r) < N(b). We get that r = a - qb and a,qb \in I, which implies that r \in I. Now we assumed that b is minimal, so this means that r = 0 since we assumed that b \neq 0. So this means that a = qb and therefore a \in (b).