In mathematics, in particular the study of abstract algebra, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains.
Let R be an integral domain and g : R → Z≥0 be a function from R to the non-negative integers.
Denote by 0R the additive identity of R. The function g is called a Dedekind–Hasse norm on R if the following three conditions are satisfied: The third condition is a slight generalisation of condition (EF1) of Euclidean functions, as defined in the Euclidean domain article.
The notion of a Dedekind–Hasse norm was developed independently by Richard Dedekind and, later, by Helmut Hasse.
They both noticed it was precisely the extra piece of structure needed to turn an integral domain into a principal ideal domain.