How to understand ideals in $F$, which is a finite commutative ring with $1$

Question

I do not fully understand about ideals in finite rings, and I have to choose the correct answer to the following:

If $F$ is a finite commutative ring with $1,$ then

(i) Each prime ideal is a maximal ideal.

(ii) $F$ has no nontrivial maximal ideal.

(iii) $F$ may have a prime ideal which is not maximal.

(iv) $F$ is a field.

I know that $(\mathbb{Z}_6,+,.)$ is a finite commutative ring with $1$ which is not a field so (iv) is out. I have no idea about other options as I have never seen an ideal of a finite ring. Thank you for your help.

Answer 1

Your example has nontrivial maximal ideals, so (ii) is also out.

Now recall that an ideal $I$ of the unital ring $F$ is prime if and only if $F/I$ is a domain, and maximal if and only if $F/I$ is a field. Now apply something you should know about finite domains.

Answer 2

Hints:

1) A finite integral domain is a field ;

2) Fields have only one ideals which is maximal: the trivial one $\,\{0\}\,$ ;

(3) Re-read (1)