# How Does the Correspondence Theorem for Rings Prove Maximal Ideals?

• MHB
• Math Amateur
In summary: Therefore, $B$ consists of two elements: $B=\{ \phi (I), \phi (R) \}$. We have $\phi (I) = (0)$ and $\phi (R) = R/I$. Thus $B=\{ (0), R/I \}$.$B$ is the set of ideals in $R/I$, so $R/I$ has no other ideals than $(0)$ and $R/I$.Conversely, $R/I$ has no other ideals than $(0)$ and $R/I$, i.e., $B=\{ (0), R/I \} Math Amateur Gold Member MHB 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: 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 ============================================================ 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 Peter said: 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 $$\displaystyle I$$ 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 ============================================================ 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: 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: $$\displaystyle I$$ maximal $$\displaystyle \Longrightarrow$$ there are no ideals in $$\displaystyle R$$ that contain $$\displaystyle I$$ except $$\displaystyle R$$ itself ... $$\displaystyle \Longrightarrow$$ there are no ideals in $$\displaystyle R/I$$ (except $$\displaystyle R/I$$ itself) since there exists a bijection between the set of ideals of $$\displaystyle R/I$$ and the ideals of $$\displaystyle R$$ containing $$\displaystyle I$$ ... ... BUT ... it seems that the only ideal in $$\displaystyle R/I$$ is $$\displaystyle R/I$$ itself ... but how do we explain the existence of $$\displaystyle (0)$$ ...? Seems that I still need some help ... ... Peter Given$I\lhd R$(notation:$I$is ideal in$R$),$I$is proper, i.e.,$I\neq (0)$and$I\neq R$. If$J\lhd R$, define$\overline J = \{a+I \mid a\in J \}$, you can prove that$\overline J = J/I$. Define$A= \{ J\lhd R \mid I\subset J \}$and$B=\{ K \lhd R/I \}$. The Correspondence Theorem says that there is a bijection$\phi : A\to B$given by$J\mapsto \overline J=J/I$. What does this say? a) If we have an ideal$K\lhd R/I$then there exists an ideal$J\lhd R$with$I\subset J$and$J/I=K$b) If we have an ideal$J\lhd R$such that$I\subset J$then$J/I \lhd R/I$Let$I\lhd R$be maximal, then$I$is proper and there are no ideals between$I$and$R$. This means that$A$consists of two elements:$A= \{ I, R \}$. Therefore,$B$consists of two elements:$B=\{ \phi (I), \phi (R) \}$. We have$\phi (I) = (0)$and$\phi (R) = R/I$. Thus$B=\{ (0), R/I \}$.$B$is the set of ideals in$R/I$, so$R/I$has no other ideals than$(0)$and$R/I$. Conversely,$R/I$has no other ideals than$(0)$and$R/I$, i.e.,$B=\{ (0), R/I \}$. Then$A= \{ \phi ^{-1} ((0)), \phi ^{-1} (R/I) \} = \{ I, R \}$Can you fill in the the details and apply example 5.8, now? I am going to have a break. If necessary, we continue later. steenis said: Given$I\lhd R$(notation:$I$is ideal in$R$),$I$is proper, i.e.,$I\neq (0)$and$I\neq R$. If$J\lhd R$, define$\overline J = \{a+I \mid a\in J \}$, you can prove that$\overline J = J/I$. Define$A= \{ J\lhd R \mid I\subset J \}$and$B=\{ K \lhd R/I \}$. The Correspondence Theorem says that there is a bijection$\phi : A\to B$given by$J\mapsto \overline J=J/I$. What does this say? a) If we have an ideal$K\lhd R/I$then there exists an ideal$J\lhd R$with$I\subset J$and$J/I=K$b) If we have an ideal$J\lhd R$such that$I\subset J$then$J/I \lhd R/I$Let$I\lhd R$be maximal, then$I$is proper and there are no ideals between$I$and$R$. This means that$A$consists of two elements:$A= \{ I, R \}$. Therefore,$B$consists of two elements:$B=\{ \phi (I), \phi (R) \}$. We have$\phi (I) = (0)$and$\phi (R) = R/I$. Thus$B=\{ (0), R/I \}$.$B$is the set of ideals in$R/I$, so$R/I$has no other ideals than$(0)$and$R/I$. Conversely,$R/I$has no other ideals than$(0)$and$R/I$, i.e.,$B=\{ (0), R/I \}$. Then$A= \{ \phi ^{-1} ((0)), \phi ^{-1} (R/I) \} = \{ I, R \}\$

Can you fill in the the details and apply example 5.8, now? I am going to have a break. If necessary, we continue later.

Reflecting on what you have said ...

Thanks again,

Peter

## 1. What is the definition of a maximal ideal in a ring?

A maximal ideal in a ring is a proper, non-zero ideal that is not contained in any other proper, non-zero ideal. This means that there are no other ideals in the ring that properly contain the maximal ideal.

## 2. How does the Correspondence Theorem for rings relate to maximal ideals?

The Correspondence Theorem states that there is a bijective correspondence between the set of all ideals in a ring and the set of all ideals in the quotient ring. This means that for every ideal in the original ring, there is a unique ideal in the quotient ring that corresponds to it. This also means that maximal ideals in the original ring correspond to maximal ideals in the quotient ring.

## 3. Can a ring have more than one maximal ideal?

Yes, a ring can have multiple maximal ideals. In fact, if a ring has at least two maximal ideals, then it is not a field.

## 4. How is Proposition 5.9 used in the context of maximal ideals and the Correspondence Theorem?

Proposition 5.9 states that if I is an ideal in a ring R, then the quotient ring R/I is a field if and only if I is a maximal ideal. This proposition is useful in proving that certain ideals are maximal, and therefore correspond to fields in the quotient ring.

## 5. Are maximal ideals unique in a ring?

No, maximal ideals are not necessarily unique in a ring. In fact, in some rings there may not be any maximal ideals at all. However, if a ring has at least one maximal ideal, then it will have infinitely many maximal ideals.

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