MHB Definition of a Euclidean Domain ....

Math Amateur
Gold Member
MHB
Messages
3,920
Reaction score
48
In the book "The Basics of Abstract Algebra" Bland defines a Euclidean Domain using two conditions as follows:View attachment 8256
View attachment 8257In 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:View attachment 8258What 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
 
Physics news on Phys.org
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.
 
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
Back
Top