Math 2.0

Commutative algebra Theorem

Euclidean domains are principal ideal domains

Builds on Euclidean domainPrincipal ideal domain

Statement

Let RR be a Euclidean domain. Then RR is a principal ideal domain.

Remarks

The proof: let I≠0I \neq 0 be an ideal and let b∈Ib \in I be a nonzero element with δ(b)\delta(b) minimal. For a∈Ia \in I, write a=bq+ra = b q + r. Then r∈Ir \in I and δ(r)<δ(b)\delta(r) < \delta(b) is impossible, so r=0r = 0 and I=(b)I = (b). The converse is false: ℤ[1+−192]\mathbb{Z}[\tfrac{1 + \sqrt{-19}}{2}] is a principal ideal domain that is not Euclidean.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionIdeal, definition, no evidence levelIdealdefinitionZero divisor, definition, no evidence levelZero divisordefinitionIdeal generated by a subset, definition, no evidence levelIdeal generated by a …definitionIntegral domain, definition, no evidence levelIntegral domaindefinitionPrincipal ideal, definition, no evidence levelPrincipal idealdefinitionEuclidean domain, definition, no evidence levelEuclidean domaindefinitionPrincipal ideal domain, definition, no evidence levelPrincipal ideal domaindefinitionEuclidean domains are principal ideal domains, statement, no evidence levelEuclidean domains are…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Ideal definition
  5. Zero divisor definition
  6. Ideal generated by a subset definition
  7. Integral domain definition
  8. Principal ideal definition
  9. Euclidean domain definition
  10. Principal ideal domain definition
  11. Euclidean domains are principal ideal domains statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Principal_ideal_domain, Properties
About this page
Tag0QKVE1GD9A
URIm2://0QKVE1GD9A
Kindstatement, theorem
WritingW0 stub
MSC13F07
StatusThis copy of the site has no build data. Run m2 status 0QKVE1GD9A.
UsesEuclidean domain, Principal ideal domain

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.