Dedekind-Hasse norm

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A Dedekind-Hasse norm on a commutative unital ring R is a function N from the nonzero elements of R to the set of nonnegative integers, satisfying the following condition:

Whenever a,b \in R are both nonzero, then one of these cases holds:

  • a is an element of the ideal (b). In other words, b | a.
  • There is a nonzero element in the ideal (a,b) whose norm is strictly smaller than that of b.

Relation with other properties

Stronger properties