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The ring = of algebraic integers in a number field K is Noetherian, integrally closed, and of dimension one: to see the last property, observe that for any nonzero prime ideal I of R, R/I is a finite set, and recall that a finite integral domain is a field; so by (DD4) R is a Dedekind domain. As above, this includes all the examples considered ...
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 ...
Dedekind 1. Richard Dedekind 2. A Dedekind-infinite set is a set that can be put into a one-to-one correspondence with one of its proper subsets, indicating a type of infinity; a Dedekind-finite set is a set that is not Dedekind-infinite. (These are also spelled without the hyphen, as "Dedekind finite" and "Dedekind infinite".) def
In mathematics, an arithmetic surface over a Dedekind domain R with fraction field is a geometric object having one conventional dimension, and one other dimension provided by the infinitude of the primes. When R is the ring of integers Z, this intuition depends on the prime ideal spectrum Spec(Z) being seen as analogous to a line.
When a vector space over a field F has a finite generating set, then one may extract from it a basis consisting of a finite number n of vectors, and the space is therefore isomorphic to F n. The corresponding statement with F generalized to a principal ideal domain R is no longer true, since a basis for a finitely generated module over R might ...
Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group () ().
In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. [1] The Dedekind number M ( n ) {\displaystyle M(n)} is the number of monotone Boolean functions of n {\displaystyle n} variables.