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I need some help with understanding the proof of Proposition 5.9 ... ...Proposition 5.9 reads as follows:

View attachment 5934 In the proof of Proposition 5.9, Rotman writes:

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

Hope that someone can help ...

Peter

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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:View attachment 5936