Find all maximal ideals in Z 8

[STEMucator]

Find all maximal ideals in Z₈...

Homework Statement

http://gyazo.com/e292522bc3d99584d5abb55826b4a50f

Homework Equations

Some definitions.

http://gyazo.com/f095ef61ecc9806c8f6a95fa99dad6fb

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

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?

 

[Dick]

Homework Statement

http://gyazo.com/e292522bc3d99584d5abb55826b4a50f

Homework Equations

Some definitions.

http://gyazo.com/f095ef61ecc9806c8f6a95fa99dad6fb

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

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?

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?

 

[STEMucator]

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?

Hmm... The subgroups of Z_n 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 Z_n come from the ideals of Z that contain nZ, any maximal ideal of Z_n is of the form pZ_n where p is a prime that divides n.

 

[Dick]

Hmm... The subgroups of Z_n 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 Z_n come from the ideals of Z that contain nZ, any maximal ideal of Z_n is of the form pZ_n where p is a prime that divides n.

Right.