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A is a GCD domain satisfying ACCP. A is a Schreier domain, [6] and atomic. A is a pre-Schreier domain and atomic. A has a divisor theory in which every divisor is principal. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.) A is a Krull domain and every prime ideal of height 1 ...
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors.
The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain. Let A be an integral domain, the following are equivalent. A is a PID. Every prime ideal of A is principal. [13] A is a Dedekind domain that is a UFD.
The class number of a number field is by definition the order of the ideal class group of its ring of integers. Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus a unique factorization domain). The fundamental theorem of arithmetic says that Q has class number 1.
In the case of coefficients in a unique factorization domain R, "rational numbers" must be replaced by "field of fractions of R". This implies that, if R is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over R is a unique factorization domain. Another ...
If is an integral domain, then is an irreducible element of if and only if, for all ,, the equation = implies that the ideal generated by is equal to the ideal generated by or equal to the ideal generated by . This equivalence does not hold for general commutative rings, which is why the assumption of the ring having no nonzero zero divisors is ...
In mathematics, the ideal class group (or class group) of an algebraic number field K is the quotient group J K /P K where J K is the group of fractional ideals of the ring of integers of K, and P K is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of K.
The unique factorization property means that a non-zero non-unit r can be represented as a product of prime elements r = p 1 p 2 ⋯ p n {\displaystyle r=p_{1}p_{2}\cdots p_{n}} Then r is square-free if and only if the primes p i are pairwise non-associated (i.e. that it doesn't have two of the same prime as factors, which would make it ...