$R$ euclidean domain, show $\tilde{N}(a)=\min_{0\neq x\in\left}N(x)$ is a norm

Question

Let $R$ be an euclidean domain with a norm $N$. We define $\tilde{N}(a) = \min_{0\neq x\in\left<a\right>}N(x)$. Show that

1. $\forall a,b\neq 0\quad \tilde{N}(ab) \geq \tilde{N}(a)$
2. $\tilde{N}(a)=\min_{0\neq x\in\left<a\right>}N(x)$ is a norm

I proved the first claim, but I'm struggling with the second one and would be thankful for some hint.