Proposition

Prove that $\{ 0 \}$ and $k$ are the only ideals of a field $k$.

Solution

Let $I \ne \{ 0 \}$ be an ideal of $k$. Such an ideal must exist since $k$ is an ideal of $k$. Let $x \in I$ such that $x \ne 0$. Since $k$ is a field, $x^{-1} \in k$. Thus $1 = x^{-1}x \in I$. Since $I$ is an ideal, $\forall y \in k, y = y \cdot 1 \in I$. Thus $I = k$.