1
$\begingroup$

Find all prime ideals and all maximal ideals of $\mathbb{Z}_6$. I am confused as to go about finding them. I know all maximal ideals are prime, but not sure what to do. Thanks

$\endgroup$
  • $\begingroup$ Try finding all the ideals of $\Bbb Z_6$ and then check which ones are prime and which ones are maximal. Hint: the ideals of $\Bbb Z/6\Bbb Z$ are in correspondence with the ideals of $\Bbb Z$ containing $6\Bbb Z$. $\endgroup$ – Brent Kerby Feb 25 '15 at 22:58
  • $\begingroup$ So the ideals are just Z(sub6),2Z(sub6), 3Z(sub6), 4Z(sub6),5Z(sub6), and 6Z(sub6).I'm sorry not sure what to do from here again. I see Z(sub6)={0,1,2,3,4,5}, 2Z(sub6)={0,2,4}, 3Z(sub6)={0,3} $\endgroup$ – Nicole Feb 25 '15 at 23:19
  • $\begingroup$ Your initial list of ideals has some repeats; check to see which ones are the same as others. One approach from there: consider the ideals as additive subgroups, and look at their index. If a subgroup has prime index, then it is a maximal subgroup; and an ideal which is maximal as a subgroup is automatically a maximal ideal. $\endgroup$ – Brent Kerby Feb 25 '15 at 23:24
  • $\begingroup$ So, I'm left with Z(sub 6)=5(sub 6), 2Z(sub6)=4(sub 6), and 3Z(sub6). Ok does that make {0,2,4} as well as {0,3} maximal because they have index 3 and 2 $\endgroup$ – Nicole Feb 25 '15 at 23:33
  • $\begingroup$ And I just found a theorem in my book that says every finite integral domain is a field, thus the maximal and prime ideals will coincide. So assuming I was right above, that would go for both the maximal and prime ideals $\endgroup$ – Nicole Feb 25 '15 at 23:43
1
$\begingroup$

The number of ideals is only four. Try finding them, drawing the lattice and then checking which are prime (although maximal implies prime).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.