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When S is finite, its completion is also finite, and has the smallest number of elements among all finite complete lattices containing S. [ 12 ] The partially ordered set S is join-dense and meet-dense in the Dedekind–MacNeille completion; that is, every element of the completion is a join of some set of elements of S , and is also the meet ...
A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers. [1] A simple example is , the set of natural numbers.
Using the standard ZFC axioms for set theory, every Dedekind-finite set is also finite, but this implication cannot be proved in ZF (Zermelo–Fraenkel axioms without the axiom of choice) alone. The axiom of countable choice , a weak version of the axiom of choice, is sufficient to prove this equivalence.
A Dedekind domain can also be characterized in terms of homological algebra: an integral domain is a Dedekind domain if and only if it is a hereditary ring; that is, every submodule of a projective module over it is projective. Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is injective. [3]
The definition of excellent rings is quite involved, so we recall the definitions of the technical conditions it satisfies. Although it seems like a long list of conditions, most rings in practice are excellent, such as fields, polynomial rings, complete Noetherian rings, Dedekind domains over characteristic 0 (such as ), and quotient and localization rings of these rings.
However Dedekind's problem of computing the values of () remains difficult: no closed-form expression for () is known, and exact values of () have been found only for . [ 3 ] Definitions
A is a Dedekind domain that is a UFD. Every finitely generated ideal of A is principal (i.e., A is a Bézout domain) and A satisfies the ascending chain condition on principal ideals. A admits a Dedekind–Hasse norm. [14] Any Euclidean norm is a Dedekind-Hasse norm; thus, (5) shows that a Euclidean domain is a PID. (4) compares to:
In more detail, an arithmetic surface (over the Dedekind domain ) is a scheme with a morphism: with the following properties: is integral, normal, excellent, flat and of finite type over and the generic fiber is a non-singular, connected projective curve over () and for other in (),
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