# Euclideanness is quotient-closed

This article gives the statement, and possibly proof, of a commutative unital ring property (i.e., Euclidean ring) satisfying a commutative unital ring metaproperty (i.e., quotient-closed property of commutative unital rings)

View all commutative unital ring metaproperty satisfactions | View all commutative unital ring metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for commutative unital ring properties

Get more facts about Euclidean ring|Get more facts about quotient-closed property of commutative unital rings

## Statement

Suppose is a commutative unital ring that possesses a Euclidean norm . Suppose is an ideal inside . Then, is a Euclidean ring, with norm given by:

.

In other words, the norm of a coset is defined as the minimum of normso f all elements in the coset. (Note that this minimum is well-defined since it is the minimum over a nonempty subset of a well-ordered set.

## Related facts

- Minimum over principal ideal of Euclidean norm is a smaller multiplicatively monotone Euclidean norm
- Euclideanness is localization-closed

## Proof

**Given**: A commutative unital ring with Euclidean norm . An ideal of . is defined on by:

.

**To prove**: is a Euclidean norm on .

**Proof**: Suppose are two elements of with . Now, there exists with and . By the Euclidean division in we have:

where or . Going modulo , we get:

which can be rewritten as:

where or . Note that by definition, and by the choice of . Thus, we have or , which is precisely the condition for Euclidean division in .