is a proper ideal of and let
Then is also an ideal of such that which means we must have because is maximal
So if is a proper maximal ideal, then for every we must have
On the other hand if for proper ideal , and for every we have , we will show that must be maximal
Suppose it is not maximal. Then there exists some ideal such that
Let (such an element must exist as )
Then must equal
But we also have so that
And so we must have which contradicts our assumption that
Thus, our assumption that is not maximal is incorrect
So that must be maximal
Therefore, for a proper ideal, is maximal if and only if for every we have
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