SOLUTION: Prove field has no proper ideal?

Question 1149261: Prove field has no proper ideal?
Answer by ikleyn(52780) (Show Source): You can put this solution on YOUR website!
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Let F be a field and let I be a non-zero ideal in F.


Take a non-zero element x in I.

Since F is the field, there is a multiplicative inverse to x element y in F, such that xy = 1, the multiplicative unit in F.


Since I is an ideal, the element xy must belong to I.
Thus, 1 belongs to I.


Then any element "c"  of  F belongs to I,  because c = c*1.


It means that the ideal I coincides with the entire field F, i.e. IS NOT proper.

The proof is completed.

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It is one of the first statements of ideals theory in rings, which I firmly know and firmly remember from my years at the University.

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