Zero ideal is proper ideal

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Definition of proper ideal link

I saw in the a proof of theorem such that it says that zero ideal is proper ideal. But, why? Can you explain?

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2 Answers

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Here "proper" just means not equal to the whole ring. In a ring $R$, the only ideal that is not proper is the ideal $R$ itself.

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The zero ideal contains exactly one element: zero. Therefore as long as you have multiple elements, the ideal is proper.

Your confusion might come from the fact that the ideal generated by one is the entire ring. But the ideal generated by zero is just zero, since $a\cdot 0=0$ holds for all $a$.

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