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A GCD domain generalizes a unique factorization domain (UFD) to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian). GCD domains appear in the following chain of class inclusions:
A 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 Dedekind domain is a UFD if and only if its ideal class group is trivial. In this case, it is in fact a principal ideal domain. In general, for an integral domain A, the following conditions are equivalent: A is a UFD.
Most significantly, they identified the nilpotent and the idempotent elements as useful hypercomplex numbers for classifications. The Cayley–Dickson construction used involutions to generate complex numbers, quaternions, and octonions out of the real number system.
In a GCD domain (for instance in ), the operations of GCD and LCM are idempotent. In a Boolean ring, multiplication is idempotent. In a Tropical semiring, addition is idempotent. In a ring of quadratic matrices, the determinant of an idempotent matrix is either 0 or 1.
The following is a chain of class inclusions that describes the relationship between rings, domains and fields: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
In mathematics the Karoubi envelope (or Cauchy completion or idempotent completion) of a category C is a classification of the idempotents of C, by means of an auxiliary category. Taking the Karoubi envelope of a preadditive category gives a pseudo-abelian category , hence the construction is sometimes called the pseudo-abelian completion.
The Euclidean domains and the UFD's are subclasses of the GCD domains, domains in which a greatest common divisor of two numbers always exists. [153] In other words, a greatest common divisor may exist (for all pairs of elements in a domain), although it may not be possible to find it using a Euclidean algorithm.
However, if R is a unique factorization domain or any other GCD domain, then any two elements have a GCD. If R is a Euclidean domain in which euclidean division is given algorithmically (as is the case for instance when R = F [ X ] where F is a field , or when R is the ring of Gaussian integers ), then greatest common divisors can be computed ...