Dedekind–Hasse Norm
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. Definition Let ''R'' be an integral domain and ''g'' : ''R'' → Z≥0 be a function from ''R'' to the non-negative integers. Denote by 0''R'' the additive identity of ''R''. The function ''g'' is called a ''Dedekind–Hasse norm'' on ''R'' if the following three conditions are satisfied: * ''g''(''a'') = 0 if and only if ''a'' = 0''R'', * for any nonzero elements ''a'' and ''b'' in ''R'' either: ** ''b'' divides ''a'' in ''R'', or ** there exist elements ''x'' and ''y'' in ''R'' such that 0 < ''g''(''xa'' − ''yb'') < ''g''(''b''). The third condition is a slight generalisation of condition (EF1) of Euclidean functions, as defined in the

Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]