# Minimum over principal ideal of Euclidean norm is a smaller multiplicatively monotone Euclidean norm

## Statement

Suppose is a commutative unital ring and is a Euclidean norm on -- in particular, is a Eulidean ring. We can define a new Euclidean norm on as follows (for ):

.

In other words, it is the minimum of the norms of nonzero elements in the principal ideal generated by .

is a multiplicatively monotone norm and further, for any . Thus, is a *smaller* multiplicatively monotone Euclidean norm on .

## Related facts

## Proof

**Given**: A ring with a Euclidean norm . Define, for , .

**To prove**: is Euclidean, for , and for .

**Proof**:

- 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 .