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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.
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.
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.
if the ring R is a principal ideal domain. For example, this applies to R = Z (the integers), so an abelian group is projective if and only if it is a free abelian group. The reason is that any submodule of a free module over a principal ideal domain is free. if the ring R is a local ring. This fact is the basis of the intuition of "locally ...
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
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).