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    Every field is a simple ring

    P-5GGTM

    Area
    Algebra
    Topics
    Fields, Ideals, Rings
    Status
    draft
    • Show that every field is simple.
    Solution

    Let \(F\) be a field and let \(I\trianglelefteq F\). If \(I=(0)\) there is nothing to prove. Otherwise choose \(0\ne x\in I\). Since \(x\) is a unit, \[ 1=x^{-1}x\in I. \] Therefore every \(a\in F\) satisfies \(a=a\cdot1\in I\), so \(I=F\). Thus the only ideals of \(F\) are \((0)\) and \(F\), and every field is a simple ring.

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