Example on Maximal and prime ideal part 1

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Proof of every Maximal is prime ideal of Commutative Ring R with unity
   • Proof by definition every Maximal is prime  



for more examples:-    • Example on Maximal and prime ideal part 2  

Proof by using definition
Every Maximal ideal is prime ideal proof :-.    • Every Maximal ideal  is Prime ideal  
Maximal and Prime Ideal of Z27
First we find all the subgroups of Z27 ,,
z27 ={0,1,2,.......26}
The subgroups are those sets which are generated by each elements
cyclic generator is a element in z27 which covers all elements note these generators is neither maximal nor prime ideal here are those cyclic generators 1,2,4,5,... and all those elements which are relatively prime with 27
the remaining ideals are only generated by 3 and 9
but 9 is completely lies in 3
hence We conclude that MAXIMAL OF Z27 is 3={0,3,6,9,12,15,18,21,24}
And Every Maximal is Prime Ideal
So therefore Prime Ideal of Z27 is only 3 .
But someone say that why 9 is not a Prime Ideal
The 9 is not a prime Ideal because
As 9= {0,9,18}, By the definition of prime ideal
as 3.6 = 18 but neither 3 nor 6 belongs to 9
hence 9 is not a Prime ideal of Z27


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