Multiplicatively monotone Euclidean norm
A Euclidean norm is termed multiplicatively monotone if the norm of a nonzero product of two elements is at least equal to the norms of the elements. In symbols, if is a Euclidean norm on a commutative unital ring , we say that is multiplicatively monotone if for any such that :
Relation with other properties
- Multiplication-additive Euclidean norm: Here, the norm of a product equals the sum of the norms.
- Multiplicative Euclidean norm as long as there are no elements of norm zero.
- A multiplicatively monotone Euclidean norm takes the same value on associate elements. For full proof, refer: Multiplicatively monotone Euclidean norm is constant on associate classes
- If for in an integral domain with a multiplicatively monotone Euclidean norm, then there is no pair with , and . For full proof, refer: Multiplicatively monotone Euclidean norm admits unique Euclidean division for exact divisor
- A Euclidean norm that is both filtrative and multiplicatively monotone is a uniquely Euclidean norm. For full proof, refer: filtrative and multiplicatively monotone implies uniquely Euclidean