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The Two Ideals of Fields
A field has exactly two ideals: the zero ideal, which contains only the additive identity, and the whole field itself. These are known as trivial ideals.
Further if a commutative ring, with distinct additive and multiplicative identities, has no ideals other than the trivial ones, then it must be a field. Note that every field is also a commutative ring with distinct additive and multiplicative identities. Therefore, we can say that every field is a commutative ring with distinct additive and multiplicative identities and only trivial ideals, and vice versa.
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