Maximal Ideals and the Correspondence Theorem for Rings

[Math Amateur]

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:

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

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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:

 

[Math Amateur]

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

 

[fresh_42]

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.

 

[Math Amateur]

Thanks for the help fresh_42 ...

Just reflecting on what you have said ...

Peter