In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for non-zero elements. I am teaching an undergraduate course and I am trying to make things as simple as possible. Is there any good reason why not to define it as $0$ at $0$?



Euclidean function of Euclidean domain defined at 0
You'll find your answer and much more in the little-known paper [1] which surveys all of the dozen known ways of axiomatizing Euclidean rings (including those of Nagata and Samuel), and explores in-depth all of their logical interrelations. It's a convenient reference to have at hand when you're comparing texts which use (seemingly) different definitions of Euclidean rings / domains.


[1] Euclidean Rings. A. G. Agargun, C. R. Fletcher  

Tr. J. of Mathematics, 19, 1995, 291 - 299.  

<http://journals.tubitak.gov.tr/math/issues/mat-95-19-3/pp-291-299.pdf> 


