Ring with unique factorization for ideals (mathematics)
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.
A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields.
An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID.
In abstract algebra, a Dedekinddomain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors...
Principal ideal domains are Noetherian, they are integrally closed, they are unique factorization domains and Dedekinddomains. All Euclidean domains and all...
For example, the class group of a Dedekinddomain is trivial if and only if the ring is a unique factorization domain. Ideal class groups (or, rather,...
Richard DedekindDedekind cut DedekinddomainDedekind eta function Dedekind-infinite set Dedekind number Dedekind psi function Dedekind sum Dedekind zeta...
but a torsionfree module over a Dedekinddomain is no longer necessarily free. Torsionfree modules over a Dedekinddomain are determined (up to isomorphism)...
of integral domains and is particularly fruitful in the study of Dedekinddomains. In some sense, fractional ideals of an integral domain are like ideals...
Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given at the end). Also, a Dedekinddomain is a UFD if...
ideal domain, and not a field. R is a valuation ring with a value group isomorphic to the integers under addition. R is a local Dedekinddomain and not...
f (x) = v. The previous example K[[X]] is a special case of this. A Dedekinddomain with finitely many nonzero prime ideals P1, ..., Pn. Define f ( x )...
be defined when we replace Z by an arbitrary Dedekinddomain. (for instance, any principal-ideal domain). For instance, in control theory it can be useful...
of a Dedekinddomain, for any nonzero prime of your Dedekinddomain, there is a map from the Dedekinddomain to the quotient of the Dedekinddomain by the...
algorithm Dedekinddomain, an integral domain in which every nonzero proper ideal factors into a product of prime ideals GCD domain, an integral domain in which...
Euclidean domain. The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekinddomain. The ring...
axiom Dedekind completeness Dedekind cut Dedekind discriminant theorem DedekinddomainDedekind eta function Dedekind function Dedekind group Dedekind number...
over any unique factorization domain). A GCD domain (in particular, any Bézout domain or valuation domain). A Dedekinddomain. A symmetric algebra over a...
has a page on the topic of: Integral domainsDedekind–Hasse norm – the extra structure needed for an integral domain to be principal Zero-product property...
as for every Dedekinddomain, a ring of quadratic integers is a unique factorization domain if and only if it is a principal ideal domain. This occurs...
of the ideal ⟨ a , b ⟩ . {\displaystyle \langle a,b\rangle .} For a Dedekinddomain R , {\displaystyle R,} we may also ask, given a non-principal ideal...
a Dedekind ring (or Dedekinddomain), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers. For general Dedekind rings...
However, rings of algebraic integers satisfy the weaker property of Dedekinddomains: ideals factor uniquely into prime ideals. Factorization may also refer...
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