Minimum over principal ideal of Euclidean norm is a smaller multiplicatively monotone Euclidean norm
In other words, it is the minimum of the norms of nonzero elements in the principal ideal generated by .
- Every Euclidean ring has a unique smallest Euclidean norm
- Euclideanness is localization-closed
- Euclideanness is quotient-closed
Given: A ring with a Euclidean norm . Define, for , .
To prove: is Euclidean, for , and for .
- Proof that for all : This is direct since is the minimum over a collection of numbers that includes .
- Proof that for : This follows from the fact that the set of nonzero multiples of is contained in the set of nonzero multiples of , as well as in the set of nonzero multiples of .
- Proof that is Euclidean: Suppose with . Then, for some . Euclidean division of by with respect to the original norm yields where or . In particular, or . Thus, we have where or .