Quotient Rings .... Remarks by Adkins and Weintraub ....

In summary: Hi, Peter.In summary, the authors state that the definition of quotient rings is independent of the choice of coset representatives, meaning that if $r+I=r'+I$ and $s+I=s'+I$, then $r'=r+a$ and $s'=s+b$, where $a,b \in I$. This can be understood by considering the definitions of $r+I$ and $r'+I$ and realizing that $r'$ must be an element of both sets, thus it must be equal to $r+a$ for some $a \in I$.
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I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ...

I am currently focused on Chapter 2: Rings ...

I need help with fully understanding some remarks by Adkins and Weintraub on quotient rings on page 59 in Chapter 2 ...

The remarks by Adkins and Weintraub on quotient rings read as follows:
View attachment 7948
In the above text from A&W we read the following:

" ... ... All that needs to be checked is that this definition is independent of the choice of coset representatives. To see this suppose \(\displaystyle r + I = r' + I\) and \(\displaystyle s + I = s' + I\). Then \(\displaystyle r' = r + a\) and \(\displaystyle s' = s + b\) where \(\displaystyle a,b \in I\). ... ... ... "Can someone please (fully) explain how/why it is that \(\displaystyle r + I = r' + I\) and \(\displaystyle s + I = s' + I\) imply that \(\displaystyle r' = r + a\) and \(\displaystyle s' = s + b\) where \(\displaystyle a,b \in I\) ... ... ?
Help will be appreciated ...

Peter
 
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Hi, Peter.

Peter said:
Can someone please (fully) explain how/why it is that \(\displaystyle r + I = r' + I\) and \(\displaystyle s + I = s' + I\) imply that \(\displaystyle r' = r + a\) and \(\displaystyle s' = s + b\) where \(\displaystyle a,b \in I\) ... ... ?

Here is one possible argument:

By definition, $r + I = \{r+a: a\in I\}$ and $r' + I = \{r'+a: a\in I\}.$ Now, $r'\in r' + I$ because $0\in I$. Since $r'\in r'+I=r+I=\{r+a: a\in I\},$ $r'=r+a$ for some $a\in I$.
 
  • #3
GJA said:
Hi, Peter.
Here is one possible argument:

By definition, $r + I = \{r+a: a\in I\}$ and $r' + I = \{r'+a: a\in I\}.$ Now, $r'\in r' + I$ because $0\in I$. Since $r'\in r'+I=r+I=\{r+a: a\in I\},$ $r'=r+a$ for some $a\in I$.
Well! Thanks! Really clear ...

Appreciate your help GJA ...

Peter
 

1. What are quotient rings and how are they different from traditional rings?

A quotient ring is a mathematical concept used in abstract algebra. It is a ring that is formed by taking a larger ring and "quotienting out" or "modding out" a smaller ring within it. In simpler terms, it is like dividing a larger set into smaller, equal sets. The main difference between quotient rings and traditional rings is that in quotient rings, the elements are not necessarily numbers, but can be any mathematical object that obeys the rules of a ring.

2. How are quotient rings used in real-world applications?

Quotient rings are primarily used in abstract algebra and are fundamental to many mathematical concepts and structures. They have applications in coding theory, number theory, cryptography, and other areas of mathematics. They are also used in computer science and engineering for error correction and data compression.

3. Who are Adkins and Weintraub and what are their contributions to quotient ring theory?

Adkins and Weintraub are mathematicians who have made significant contributions to the study of quotient rings. They have co-authored several textbooks on abstract algebra, including "Algebra: An Approach via Module Theory" and "Algebra: An Approach via Geometric Viewpoints." Their work has helped to further our understanding of quotient rings and their applications.

4. Can you provide an example of a quotient ring and how it is formed?

One example of a quotient ring is the integers modulo 5, denoted as Z/5Z. This ring is formed by taking the set of integers and dividing it into 5 smaller sets based on their remainder when divided by 5. The elements of Z/5Z are {[0], [1], [2], [3], [4]}. The addition and multiplication operations are defined based on the remainder of the sum or product of two elements modulo 5. For example, [2] + [4] = [1] and [3] * [3] = [4].

5. What are some properties of quotient rings that make them useful in mathematics?

Quotient rings have several important properties that make them useful in mathematics. Firstly, they maintain the algebraic structure of the original ring, making it easier to study and manipulate. They also allow us to study the relationship between different rings and their subrings. Furthermore, they have applications in algebraic geometry, where they are used to study algebraic curves and surfaces. Lastly, quotient rings provide a powerful tool for proving theorems and solving problems in abstract algebra and other areas of mathematics.

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