1. ## Find maximal ideals

Find all the maximal ideals of :
(i) Real number IR
(ii) Integers Z
(iii) C[x]
(iv) Z_60

I got no idea how to do it, so I couldn't attempt these questions. Sorry
Can some body show me how to do it please?

Thank you very much

2. Originally Posted by knguyen2005
Find all the maximal ideals of :
(i) Real number IR
(ii) Integers Z
(iii) C[x]
(iv) Z_60

I got no idea how to do it, so I couldn't attempt these questions. Sorry
Can some body show me how to do it please?

Thank you very much
Look up your definition for ideals, then try to find ideals in the given ring first. Then you can check which ones are maximal ideals using the definition.

3. Some results you might find useful:

1) If $\displaystyle R$ is a principal ideal domain (PID) and $\displaystyle I \subseteq R$ is an ideal then $\displaystyle I$ is maximal iff $\displaystyle I$ is prime.

2) If $\displaystyle R$ is a ring then $\displaystyle <f>$ is prime iff $\displaystyle f$ is prime

4. Originally Posted by knguyen2005
Find all the maximal ideals of :
(i) Real number IR

$\displaystyle \mathbb{R}$ is a field. thus ...

(ii) Integers Z

$\displaystyle \mathbb{Z}$ is a PID. the maximal ideals are in the form $\displaystyle p\mathbb{Z},$ where $\displaystyle p$ is any prime.

(iii) C[x]

$\displaystyle \mathbb{C}[x]$ is a PID. the maximal ideals are in the form $\displaystyle <x-a>,$ where $\displaystyle a \in \mathbb{C}.$ that is because $\displaystyle \mathbb{C}$ is algebraically closed.

(iv) Z_60

let $\displaystyle \{p_1, \cdots , p_k \}$ be the set of prime divisors of $\displaystyle n> 1.$ then $\displaystyle \mathbb{Z}/n\mathbb{Z}$ has exactly $\displaystyle k$ maximal ideals: $\displaystyle m_j = p_j \mathbb{Z}/n \mathbb{Z}, \ 1 \leq j \leq k.$ (also see (ii))

I got no idea how to do it, so I couldn't attempt these questions. Sorry
Can some body show me how to do it please?

Thank you very much
see if you can get some ideas now!

5. Don't worry, I got it now, Thanks again