Maximal Ideals and the Correspondence Theorem for Rings

In summary, Rotman's book: Advanced Modern Algebra (AMA) provides a proof of the Correspondence Theorem for Rings. This theorem states that if there are no other ideals in a ring other than the zero ideal, then the ring is maximal. Rotman uses the concept of correspondence to show that if there is a bijection between the set of ideals of a ring and the set of ideals of a larger ring containing the original ring, then the original ring is a maximal ideal. However, the only ideal in the original ring is the original ring itself. Rotman notes that this can be explained by assuming that the ideal ##\overline{J}= J+I## in the ring ##R/I## exists. Since
  • #1
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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

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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:


?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
 
  • #3
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
 

1. What is a maximal ideal in a ring?

A maximal ideal in a ring is an ideal that is not contained in any other proper ideal of the ring. In other words, there are no other ideals between the maximal ideal and the ring itself. This means that the maximal ideal is as large as it can be within the ring.

2. How do you prove that an ideal is maximal?

To prove that an ideal is maximal, you need to show that it is not contained in any other proper ideal of the ring. This can be done by assuming that there is a larger ideal containing the maximal ideal, and then showing that this leads to a contradiction. Alternatively, you can use the definition of a maximal ideal and show that it satisfies all the necessary conditions.

3. What is the Correspondence Theorem for Rings?

The Correspondence Theorem for Rings states that there is a one-to-one correspondence between the set of ideals of a ring and the set of all its quotient rings. In other words, for every ideal of a ring, there is a unique quotient ring that corresponds to it, and vice versa.

4. How do you use the Correspondence Theorem to simplify ring operations?

The Correspondence Theorem can be used to simplify ring operations by allowing us to work with ideals instead of elements of the ring. This is because the quotient ring corresponding to an ideal has the same structure as the ideal itself. Therefore, operations on the quotient ring can be translated back to operations on the original ring, making calculations and proofs easier.

5. Can the Correspondence Theorem be extended to other algebraic structures?

Yes, the Correspondence Theorem can be extended to other algebraic structures such as groups and modules. In these cases, the correspondence is between subgroups and submodules, respectively, and their corresponding quotient structures. The same principles and properties apply, making the Correspondence Theorem a powerful tool in studying these structures.

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