Maximal Ideals and the Correspondence Theorem for Rings

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SUMMARY

The discussion centers on the Correspondence Theorem for Rings as presented in Joseph J. Rotman's "Advanced Modern Algebra." Specifically, Proposition 5.9 states that an ideal \( I \) is maximal if and only if the quotient ring \( R/I \) contains no ideals other than \( (0) \) and \( R/I \) itself. The proof hinges on the bijection between the ideals of \( R/I \) and the ideals of \( R \) that contain \( I \), leading to the conclusion that if \( I \) is maximal, the only ideal in \( R/I \) is \( R/I \) itself, with \( (0) \) being identified as \( I/I \).

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I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.1 Prime Ideals and Maximal Ideals ...

I need some help with understanding the proof of Proposition 5.9 ... ...Proposition 5.9 reads as follows:
?temp_hash=a8c6b30ae57a4ffe62159d453fb0011e.png
In the proof of Proposition 5.9, Rotman writes:

" ... ... The Correspondence Theorem for Rings shows that ##I## is a maximal ideal if and only if ##R/I## has no ideals other than ##(0)## and ##R/I## itself ... ... "

My question is: how exactly (in clear and simple terms) does Rotman's statement of the Correspondence Theorem for Rings lead to the statement that "##I## is a maximal ideal if and only if ##R/I## has no ideals other than ##(0)## and ##R/I## itself" ... ...

Hope that someone can help ...

Peter

============================================================

The above post refers to Rotman's statement of the Correspondence Theorem for Rings, so I am providing a statement of that theorem and its proof, as follows:


?temp_hash=a8c6b30ae57a4ffe62159d453fb0011e.png
 

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Maybe I should not be responding to my own post but I have been reflecting on the question in the above post and now suspect that the answer is quite simple and goes along the lines ... ... as follows:

##I## maximal

##\Longrightarrow## there are no ideals in ##R## that contain ##I## except ##R## itself ...

##\Longrightarrow## there are no ideals in ##R/I## (except ##R/I##) since there exists a bijection between the set of ideals of ##R/I## and the ideals of ##R## containing ##I## ... ...

BUT ... it seems that the only ideal in ##R/I## is ##R/I## itself ... but how do we explain the existence of (0) ...

Seems that I still need some help ... ...

Peter
 
Math Amateur said:
I maximal

##\Longrightarrow## there are no ideals in ##R## that contain ##I## except ##R## itself ...
... and ##I##.

##\Longrightarrow## there are no ideals in ##R/I## (except ##R/I##
... and ##I/I##

) since there exists a bijection between the set of ideals of ##R/I## and the ideals of ##R## containing ##I## ... ...

BUT ... it seems that the only ideal in ##R/I## is ##R/I## itself ...
... and ##I/I##
but how do we explain the existence of (0) ...
We do by identifying ##I/I = (0)##.

You could also assume an ideal ##\overline{J}= J+I## in ##R/I##. Since ##\overline{0} = 0+I \in \overline{J}##, this means ##I \subseteq J## which means by maximality of ##I## that ##J \in \{I\,,\,R\}##.
And this holds the other way round, too.
 
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Thanks for the help fresh_42 ...

Just reflecting on what you have said ...

Peter
 

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