Factorizaton in an Euclidean ring

Question

I have a doubt concerning Lemma 3.7.4 from Topics in Algebra by I. N. Herstein.

The statement of the Lemma is:

Let $R$ be a Euclidean ring. Then every element in $R$ is either a unit in $R$ or can be written as the product of a finite number of prime elements of $R$.

My question is that $0$ is an element of $R$ but it is neither a unit nor a product of finitely many prime elements of $R$. So is there a typo and the Lemma holds only for all nonzero elements of $R$?

Answer 1

You seem to be right. Herstein's definition of a prime element on page 147 definitely implies that $0$ is not prime, and his proof by induction on $d(a)$ fails to account for the case $a = 0$ because he intentionally leaves $d(0)$ undefined.