Definition of a Euclidean Domain ....

[Math Amateur]

In the book "The Basics of Abstract Algebra" Bland defines a Euclidean Domain using two conditions as follows:


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In the book "Abstract Algebra"by Dummit and Foote we find that a Euclidean Domain is defined using only one of Bland's conditions ... as follows:


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What are the consequences of these different definitions ... for example does D&F's definition allow some structures to be Euclidean Domains that are not recognized as such under Bland's definition ...


Peter

 

[steenis]

I do not knoe the answer to this question.
You can do a search in Google and start with reading the Wikipedia article

https://en.wikipedia.org/wiki/Euclidean_domain

After that, for instance read:

http://www.math.uconn.edu/~kconrad/blurbs/ringtheory/euclideanrk.pdf
https://www.math.hmc.edu/~davis/171hw/171lec119.pdf

and there is much more.
Maybe someone else can help you more directly.

 

[Olinguito]

Let $N$ be the norm of the integral domain $R$ in F&D’s definition. Define another norm $N^\ast$ by
$$N^\ast(a)\ =\ \min_{b\in R\setminus\left\{0_R\right\}}N(ab).$$
Then $R$ is still a Euclidean domain with norm $N^\ast$ and $N^\ast$ is also a Euclidean valuation in Bland’s definition.

The most important property of a Euclidean domain is the second one in Bland; many results about Euclidean domains involve only the second property but not the first. IMHO the importance of the first property only shows up in the study of ideals and algebraic-number theory.