Euclidean Rings - Rotman Example 3.76

In summary, Rotman's book discusses how to define a Euclidean ring, and how to satisfy the two points of the definition. However, he does not explain what he means by "point (ii)."
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I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ...

I am currently focused on Section 3.5 From Numbers to Polynomials ...

I need help with Example 3.76 ... ... the example concerns Euclidean rings and their defining characteristics so I am including the definition of a Euclidean ring in the relevant text shown below ... ...

The relevant text from Rotman's book is as follows:https://www.physicsforums.com/attachments/4649

View attachment 4648
I am trying to understand Example 3.76 which indicates that every field is a Euclidean ring ... ...

I can see that point (i) of the definition is satisfied with \(\displaystyle \partial\) set identically to zero ...... BUT ... I fail to understand what Rotman is saying about how point (ii) is satisfied ... ...

In order for (ii) to be satisfied, for every \(\displaystyle g \in R\) and every \(\displaystyle f \in R^{\times}\) we have to find \(\displaystyle q, r \in R\) such that:

\(\displaystyle g = qf + r\) ... ... ... (*)

... BUT ...

Rotman says to set \(\displaystyle q = f^{-1}\) and \(\displaystyle r = 0\)

but if we do this (*) above becomes

\(\displaystyle g = f f^{-1} + 0 = 1\) ...

but \(\displaystyle g\) may be any element of \(\displaystyle R\) ... ?Can someone please explain what is going on ... that is, what Rotman means in this example ...

Hope someone can help ...

Peter
 
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Peter said:
Rotman says to set \(\displaystyle q = f^{-1}\) and \(\displaystyle r = 0\) but if we do this (*) above becomes \(\displaystyle g = f f^{-1} + 0 = 1\) ... but \(\displaystyle g\) may be any element of \(\displaystyle R\) ... ?
You are right. The book should say: if $g\in R$ and $f\in R^{\times}$ set $q=gf^{-1}$ and $r=0.$ So, for all $g\in R$ and for all $f\in R^{\times}$ we verify $g=(\underbrace{gf^{-1}}_{q})\;f+\underbrace{0}_{r}.$
 
  • #3
Fernando Revilla said:
You are right. The book should say: if $g\in R$ and $f\in R^{\times}$ set $q=gf^{-1}$ and $r=0.$ So, for all $g\in R$ and for all $f\in R^{\times}$ we verify $g=(\underbrace{gf^{-1}}_{q})\;f+\underbrace{0}_{r}.$

Thanks for that clarification Fernando ... I appreciate your help ...

Peter
 

FAQ: Euclidean Rings - Rotman Example 3.76

What is an Euclidean Ring?

An Euclidean Ring is a mathematical structure that combines the properties of both a ring and an Euclidean domain. It is a set of elements with two binary operations, addition and multiplication, that follow certain rules such as commutativity, associativity, and distributivity. In addition, it also has a function called the Euclidean function, which measures the "size" of an element and allows for division with remainder.

What is the Euclidean function used for in Euclidean Rings?

The Euclidean function is used to measure the "size" of an element in an Euclidean Ring. It assigns a non-negative integer value to each non-zero element, with the property that the size of the sum of two elements is less than or equal to the maximum of their individual sizes. This function is also used to determine the quotient and remainder when dividing two elements in an Euclidean Ring.

How is the Euclidean function defined in an Euclidean Ring?

The Euclidean function in an Euclidean Ring is typically defined as a mapping from the set of non-zero elements to the set of non-negative integers, satisfying the following properties:

  • The size of any non-zero element is a non-negative integer.
  • The size of the sum of two elements is less than or equal to the maximum of their individual sizes.
  • The size of any non-zero element is strictly less than the size of its product with any non-zero element.

What is an example of an Euclidean Ring?

An example of an Euclidean Ring is the set of integers, denoted by Z. The Euclidean function in this ring is simply the absolute value function, which assigns the absolute value of an integer as its size. This ring satisfies all the properties of an Euclidean Ring, making it a commonly used example in mathematics.

How are Euclidean Rings different from Euclidean domains?

Euclidean Rings and Euclidean domains have similar properties, but the main difference is that Euclidean Rings have the additional structure of a ring, while Euclidean domains do not. This means that in Euclidean Rings, we can perform operations like addition and multiplication, while in Euclidean domains, we can only perform division with remainder. Additionally, Euclidean Rings may not have unique factorization, while Euclidean domains always have unique factorization for their elements.

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