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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 $$r + I = r' + I$$ and $$s + I = s' + I$$. Then $$r' = r + a$$ and $$s' = s + b$$ where $$a,b \in I$$. ... ... ... "Can someone please (fully) explain how/why it is that $$r + I = r' + I$$ and $$s + I = s' + I$$ imply that $$r' = r + a$$ and $$s' = s + b$$ where $$a,b \in I$$ ... ... ?
Help will be appreciated ...
Peter
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 $$r + I = r' + I$$ and $$s + I = s' + I$$. Then $$r' = r + a$$ and $$s' = s + b$$ where $$a,b \in I$$. ... ... ... "Can someone please (fully) explain how/why it is that $$r + I = r' + I$$ and $$s + I = s' + I$$ imply that $$r' = r + a$$ and $$s' = s + b$$ where $$a,b \in I$$ ... ... ?
Help will be appreciated ...
Peter