And if it is, how do you define division and an Euclidean function?
(The question was put on hold asking for context information, definitions, motivation, etc; and in the meantime I found the answer to my question, but I'll write it down here for completeness.)
The most basic example is $\Bbb Z$, that is an Euclidean ring with the absolute value as Euclidean function: For $a,b \in \Bbb Z$ with $b \neq 0$, there exist $q,r \in \Bbb Z$ with $a = bq + r$ and $|r| < |b|$.
Now take $\Bbb Z \times \Bbb Z$. Is it an Euclidean ring? What would be an Euclidean function for this ring?
Update: I was trying to define an Euclidean function with values in the natural numbers, and that works for integral domains but doesn't work for more general rings with zero-divisors. For example, $\Bbb Z \times \Bbb Z$ has no Euclidean function $\varphi: \Bbb Z \times \Bbb Z \rightarrow \Bbb N$ (Fletcher [1]).
Samuel [2] defines an Euclidean ring $R$ with a map $\varphi:R \rightarrow W$, $(W,\leq)$ a well-ordered set, such that for $a,b \in R$ with $b \neq 0$, there exist $q,r \in R$ with $a = bq + r$ and $\varphi(r) < \varphi(b)$, and shows that a product of Euclidean rings is Euclidean in this sense. And this answers my question.
The notion of Euclidean rings that are not necessarily domains is useful because some algorithms can be generalized to work over general Euclidean rings, even with zero-divisors. For example, Magma considers $\Bbb Z/m\Bbb Z$ an Euclidean ring, and in particular $\Bbb Z/p^k\Bbb Z$, as well as other Galois rings, are all Euclidean rings, and it can compute Groebner bases in polynomial rings over these rings. I'm implementing algorithms for commutative algebra and I want to compute with $R$-modules as general as possible. I'm looking at generalizations of Buchberger's, HNF, etc, and I want more examples of rings where I can solve linear systems. That's what motivated my original question.
[1] C.R. Fletcher, Euclidean rings, J. London Math. Soc. 41 (1971), 79-82
[2] P. Samuel, About Euclidean rings, J. Algebra 19 (1971), 282-301