Ok, I am new to the concepts of maximal ideals and prime ideals. I know the definitions for both, but I am kind of stuck with understanding the examples. So, any help would be much appreciated.
Consider the ring $\mathbb{Z}$, which is commutative with unity. Let $I_1 = (0), I_2 =(−9), I_3 = (23)$ and $I_4 = (28)$
- List all $I_i$ that are prime ideals.
- List all $I_i$ that are maximal ideals.
- For each $I_i$ that is not a prime ideal, find $a_i, b_i \in \mathbb{Z}\setminus I_i$ such that $a_ib_i \in I_i$.
- For each $I_i$ that is not a maximal ideal, find a maximal ideal $M_i$ such that $I_i \subset M_i$.
My attempt:
- $I_1 = (0)$ and $I_3 = (23)$ are the prime ideals.
- $I_3 = (23)$ is the maximal ideal. I know $0$ is not a maximal ideal, but I exactly don't understand why.
- Still confused here. Does it mean like this: Since, $I_2=(-9)$ is not a prime ideal, then we can find $a_2 = -3$ and $b_2 = 3$, so that $a_2b_2 = (-9)$ and for $I_4 = 28, a_4 = 4$ and $b_4 = 7$(Again I am not exactly sure here).
- Confused.
Best Answer
Your responses for the first three are exactly correct. To see why $\langle 0 \rangle$ is not maximal in $\mathbb{Z}$, notice that $\langle 0 \rangle \subsetneq \langle x \rangle$ for any nonzero $x \in \mathbb{Z}$.
For the fourth, the definition of a maximal ideal $I$ is that there are no other ideals $J$ such that $I \subsetneq J \subsetneq R$. For example, $\langle 4 \rangle$ is not maximal since $\langle 4 \rangle \subsetneq \langle 2 \rangle \subsetneq R$. If you can convince yourself that this is true, then you should be able to proceed with your problem.
As an aside, there are some very useful facts you should be aware of (and prove, if possible):