# Find all maximal ideals in Z 8

1. Jan 28, 2013

### Zondrina

Find all maximal ideals in Z8...

1. The problem statement, all variables and given/known data

http://gyazo.com/e292522bc3d99584d5abb55826b4a50f

2. Relevant equations

Some definitions.

I was also thinking about using a lattice of ideals to show this.

3. The attempt at a solution

Okay, when I draw out the ideal lattice for these, it's obvious to see which ideals are maximal. For part a, b, and c.

a) For Z8, <2> is maximal.
b) For Z10, <2> and <5> are maximal.
c) For Z12, <2> and <3> are maximal.
d) Requires a proof. Lattice fails.

My question is, how would I argue this without the aid of a lattice for a, b and c?

2. Jan 28, 2013

### Dick

Re: Find all maximal ideals in Z8...

An ideal must be a subgroup. What do the subgroups of Z_n look like? Think about prime divisors of n. Can you characterize a maximal ideal in terms of those?

3. Jan 28, 2013

### Zondrina

Re: Find all maximal ideals in Z8...

Hmm... The subgroups of Zn are cyclic and the order of each subgroup divides the order of the group.

In each case then, ideals of the form <p> where p is a prime that divides the order of the group are maximal ideals?

EDIT : So I suppose I could conclude since all ideals of Zn come from the ideals of Z that contain nZ, any maximal ideal of Zn is of the form pZn where p is a prime that divides n.

Last edited: Jan 28, 2013
4. Jan 28, 2013

### Dick

Re: Find all maximal ideals in Z8...

Right.