101.15 An elementary proof that not all principal ideal domains are Euclidean domains
About: This article is published in The Mathematical Gazette.The article was published on 2017-07-01 and is currently open access. It has received None citation(s) till now. The article focuses on the topic(s): Principal ideal & Elementary proof.
Summary (1 min read)
- A standard result in undergraduate algebra courses is that every Euclidean domain (ED) is a principal ideal domain (PID).
- It is routinely stated, but rarely proved, that the converse is false.
- A more elementary proof, accessible to advanced undergraduates, is given by Cámpoli  in 1988, though with a more restricted definition of Euclidean norm than Motzkin uses.
- Notice this does not include the property, usually included in the definition of Euclidean function, that if a|b then d(a) ≤ d(b).
4. Quasi-Euclidean domains
- The former contradicts the minimality of d(b), while the latter contradicts the assumption that a is not in bA.
- This suggests the following definition of a quasi-Euclidean domain (QED).
- (Note that this is not a standard definition, and there are other definitions of quasiEuclidean domains in the literature, not necessarily equivalent to this one.).
In this situation, A is called a quasi-Euclidean domain (or Motzkin domain).
- The above proof then generalizes immediately to a proof that every QED is a PID.
- It is not obvious that the authors have gained anything, as it is not obvious that there exist quasi-Euclidean domains that are not Euclidean.
5. R is a principal ideal domain
- The large dots represent the elements of R in the Argand diagram.
- The small dots represent all the remaining values of a/b, that is a/b = q/2.
- Adding ordinary integers to a/b as necessary, the authors may assume that a/b = (±1 + √ 19)/4, and by symmetry these two cases are essentially the same.
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Q1. What are the contributions in "An elementary proof that not all principal ideal domains are euclidean domains" ?
The first part of this proof, that R is not a Euclidean domain, Cámpoli attributes to the referee of his paper. It is worth remarking, however, that that proof, with very little modification, actually proves Motzkin ’ s slightly more general result, namely that R is not a Euclidean domain under the following definition.