# Determining prime ideals and maximal ideals

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

• 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 $6\Bbb Z$. – Brent Kerby Feb 25 '15 at 22:58
• 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} – Nicole Feb 25 '15 at 23:19
• 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. – Brent Kerby Feb 25 '15 at 23:24
• 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 – Nicole Feb 25 '15 at 23:33
• 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 – Nicole Feb 25 '15 at 23:43