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    The nilradical is contained in the Jacobson radical

    P-3EZG5

    Area
    Algebra
    Topics
    Ideals, Jacobson Radical, Nilpotence
    Status
    draft

    The nilradical is contained in the Jacobson radical, i.e. \[ \nilrad{R} \subseteq J(R) .\]

    Solution

    For a commutative ring \(R\), \(\nilrad{R}\) is the intersection of all prime ideals (E-2ZO7O) and \(J(R)\) is the intersection of all maximal ideals. Every maximal ideal is prime, so an element in every prime ideal is in every maximal ideal.

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