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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.
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.
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 converse is also true: if a prime ideal has height n, then it is a minimal prime ideal over an ideal generated by n elements. [ 1 ] The principal ideal theorem and the generalization, the height theorem, both follow from the fundamental theorem of dimension theory in commutative algebra (see also below for the direct proofs).
In algebra, the elementary divisors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If is a PID and a finitely generated-module, then M is isomorphic to a finite direct sum of the form
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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]