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Math Help - Some Radical Ideal proofs

  1. #1
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    Some Radical Ideal proofs

    If I is an Ideal for communitive ring R, prove Rad(I) is and Ideal of R and that I is an ideal of Rad(I)


    For the first, I looked at for r ∈ R, a ∈ Rad(I). Therefore a^n ∈ I for some positive integer n.

    (ra)^n = r^n*a^n ∈ I since r^n ∈ R and since I is an Ideal.

    Therefore since (ra)^n ∈ I, ra ∈ Rad(I). So Rad(I) is an Ideal of R.

    But for the second, I let a ∈ I, b ∈ Rad(I), therefore b^n ∈ I, but I don't know how to prove ba ∈ I.

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by Seda1 View Post
    If I is an Ideal for communitive ring R, prove Rad(I) is and Ideal of R and that I is an ideal of Rad(I)


    For the first, I looked at for r ∈ R, a ∈ Rad(I). Therefore a^n ∈ I for some positive integer n.

    (ra)^n = r^n*a^n ∈ I since r^n ∈ R and since I is an Ideal.

    Therefore since (ra)^n ∈ I, ra ∈ Rad(I). So Rad(I) is an Ideal of R.

    But for the second, I let a ∈ I, b ∈ Rad(I), therefore b^n ∈ I, but I don't know how to prove ba ∈ I.

    Thanks in advance!


    Why would you have to prove ba\in I? This is always true since I is an ideal...

    All you need to prove is I\subset Rad(I) (why?)

    Tonio
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