I cite from Wikipedia: 
 Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields


All of these properties have a well-known *geometric* interpretation, except perhaps "(norm) Euclidean domain". Do you know one?



geometric interpretation of "Euclidean domain"
It is a theorem of Pierre Samuel that if $R$ is normal and of finite type over a field $k$, then the following are equivalent:


1) The map taking $x$ to ($1$ plus the $k$-dimension of $R/(x)$) is a degree map that makes $R$ euclidean.


2) $R$ is a PID and there exists a fractional ideal whose divisor has odd degree at infinity.


The relevant paper is from the Journal of Algebra, 1971.


