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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.

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    $\begingroup$ Hint: what is the relationship between prime ideals, maximal ideals, integral domains, and fields? What do you know about finite integral domains? $\endgroup$ – user29743 Jun 4 '13 at 21:18
  • $\begingroup$ $\{0, 2, 4\}$ is an ideal in $\mathbb Z_6$, the image of $2\mathbb Z$ in $\mathbb Z/6\mathbb Z$. $\endgroup$ – Mark Bennet Jun 4 '13 at 21:19
  • $\begingroup$ @countinghaus $R$ be a ring and $I$ be an ideal of $R$ then $I$ is maxiaml iff $R/I$ is field, and prime iff $R/I$ is an integral domain $\endgroup$ – Marso Jun 4 '13 at 21:21
  • $\begingroup$ @TaxiDriver - I edited the question and the title - I hope this is OK. You will see from the answers that you can do the whole question using general principles. I put a comment about an ideal in the ring you cited as an example. $\mathbb Z_n$ is a source of further examples (where $n$ is composite, this is a ring, not a field). $\endgroup$ – Mark Bennet Jun 4 '13 at 21:34
  • $\begingroup$ @MarkBennet I am simply pleased and delihted $\endgroup$ – Marso Jun 4 '13 at 21:37
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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.

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    $\begingroup$ Andreassssssssssssssss Thank you $\endgroup$ – Marso Jun 4 '13 at 21:22
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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)

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  • $\begingroup$ Don thank youuuuuuuuu $\endgroup$ – Marso Jun 4 '13 at 21:23

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