Field/Two ideals/Fact/Proof

Proof

If ${}R$ is a field, then there exists the zero ideal and the unit ideal, and these are different ideals. Let ${}I$ be an ideal in ${}R$ different from ${}0$. Then ${}I$ contains some element ${}x\neq 0$, which is a unit. Therefore, ${}1=xx^{-1}\in I$ and thus ${}I=R$.

Suppose now that ${}R$ is a commutative ring with exactly two ideals. Then ${}R$ is not the zero ring. Let now ${}x$ be an element in ${}R$ different from ${}0$. The principal ideal generated by ${}x$, that is ${}Rx$, is ${}\neq 0$, and therefore it must be the other ideal, which is the unit ideal. In particular, this means ${}1\in Rx$. Hence, ${}1=xr$ for some ${}r\in R$, so that ${}x$ is a unit.

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