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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.
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 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]
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 (),
The Dedekind-Kummer theorem holds more generally than in the situation of number fields: Let be a Dedekind domain contained in its quotient field , / a finite, separable field extension with = [] for a suitable generator and the integral closure of .
Market environment and business environment are marketing terms that refer to factors and forces that affect a firm's ability to build and maintain successful customer relationships. The business environment has been defined as "the totality of physical and social factors that are taken directly into consideration in the decision-making ...
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:
any nonzero element of is contained in only a finite number of height 1 prime ideals. It is also possible to characterize Krull rings by mean of valuations only: [ 2 ] An integral domain A {\displaystyle A} is a Krull ring if there exists a family { v i } i ∈ I {\displaystyle \{v_{i}\}_{i\in I}} of discrete valuations on the field of ...