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All principal ideal domains are integrally closed. 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.
A ring in which every ideal is principal is called principal, or a principal ideal ring. A principal ideal domain (PID) is an integral domain in which every ideal is principal. Any PID is a unique factorization domain; the normal proof of unique factorization in the integers (the so-called fundamental theorem of arithmetic) holds in any PID.
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.
If D is a division ring and is a ring endomorphism which is not an automorphism, then the skew polynomial ring [,] is known to be a principal left ideal domain which is not right Noetherian, and hence it cannot be a principal right ideal ring. This shows that even for domains principal left and principal right ideal rings are different.
An integral domain A satisfies (ACCP) if and only if the polynomial ring A[t] does. [2] The analogous fact is false if A is not an integral domain. [3] An integral domain where every finitely generated ideal is principal (that is, a Bézout domain) satisfies (ACCP) if and only if it is a principal ideal domain. [4]
If A is a domain contained in a field K, we can consider the integral closure of A in K (i.e. the set of all elements of K that are integral over A). This integral closure is an integrally closed domain. Integrally closed domains also play a role in the hypothesis of the Going-down theorem.
R is a local principal 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 Dedekind domain and not a field. R is a Noetherian local domain whose maximal ideal is principal, and not a field. [1] R is an integrally closed Noetherian local ring with Krull dimension one.
The principal ideal theorem was conjectured by David Hilbert (), and was the last remaining aspect of his program on class fields to be completed, in 1929.. Emil Artin (1927, 1929) reduced the principal ideal theorem to a question about finite abelian groups: he showed that it would follow if the transfer from a finite group to its derived subgroup is trivial.