Is a Dedekind domain a PID?

Is a Dedekind domain a PID?

After the preliminaries, we prove the basic result that a local Dedekind domain is a PID. Combined with the preliminaries, this immediately gives unique factorization of ideals as products of powers of distinct primes in any Dedekind domain.

What is Dedekind Theorem?

A form of the continuity axiom for the real number system in terms of Dedekind cuts. It states that for any cut A|B of the set of real numbers there exists a real number α which is either the largest in the class A or the smallest in the class B.

What are Dedekind cuts used for?

The important purpose of the Dedekind cut is to work with number sets that are not complete. The cut itself can represent a number not in the original collection of numbers (most often rational numbers).

How did Dedekind describe continuity?

Dedekind defined “continuity” through the use of the mathematical concept known as an “infinitesimal”. He argued the “infinitesimal” is not based upon spatial or geometrical intuition.

Is a field Dedekind domain?

A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field.

What is a normal ring?

A commutative ring with identity R is called normal if it is reduced (i.e. has no nilpotent elements ≠0) and is integrally closed in its complete ring of fractions (cf. Thus, R is normal if for each prime ideal p the localization Rp is an integral domain and is closed in its field of fractions.

How do you prove a set is a Dedekind cut?

Negation: Given any set X of rational numbers, let −X denote the set of the negatives of those rational numbers. That is x ∈ X if and only if −x ∈ −X. If (A, B) is a Dedekind cut, then −(A, B) is defined to be (−B,−A). This is pretty clearly a Dedekind cut.

How do you prove something is a Dedekind cut?

The Sign: A Dedekind cut (A, B) is called positive if 0 ∈ A and nega- tive if 0 ∈ B. If (A, B) is neither positive nor negative, then (A, B) is the cut representing 0. If (A, B) is positive, then −(A, B) is negative. Likewise, if (A, B) is negative, then −(A, B) is positive.

What is concept of continuity?

continuity, in mathematics, rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. Continuity of a function is sometimes expressed by saying that if the x-values are close together, then the y-values of the function will also be close.

Is a principal ideal domain?

In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. All Euclidean domains and all fields are principal ideal domains.

Is an integral domain?

An integral domain is a nonzero commutative ring with no nonzero zero divisors. An integral domain is a commutative ring in which the zero ideal {0} is a prime ideal. Elements r with this property are called regular, so it is equivalent to require that every nonzero element of the ring be regular.