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This is not true in general. Consider the ring $R=\mathbb{Q}[x,x^{1/2},x^{1/4},x^{1/8},\dots]$ as a subring of the algebraic closure of $\mathbb{Q}[x]$, and let $I$ be the ideal $\langle x,x^{1/2},\... • 10.9k 3 votes Accepted ### Why are$(0)$and$(1)$the only ideals in a field? If$I$is an ideal in a field$K$that contains some nonzero element$x$then (since it's an ideal)$1 = xx^{-1} \in I$so$I = K\$.
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