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In principal ideal domains a near converse holds: every nonzero prime ideal is maximal. 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.
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 following result gives a complete classification of principal rings in terms of special principal rings and principal ideal domains. Zariski–Samuel theorem : Let R be a principal ring. Then R can be written as a direct product ∏ i = 1 n R i {\displaystyle \prod _{i=1}^{n}R_{i}} , where each R i is either a principal ideal domain or a ...
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.
For example, if R is a principal ideal domain, then Pic(R) vanishes. In algebraic number theory, R will be taken to be the ring of integers , which is Dedekind and thus regular. It follows that Pic( R ) is a finite group ( finiteness of class number ) that measures the deviation of the ring of integers from being a PID.
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]
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.
In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization.