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Math Help - prime and jacobson radical, please help

  1. #1
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    prime and jacobson radical, please help

    I want to find prime and jacobson radical radicals in Z[T]/(T^3), here Z = integers.
    My definition of jacobson radical is that it is intersection of maximal ideals and prime radical is intersection of prime ideals.



    Please help! Thanks
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  2. #2
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    Quote Originally Posted by peteryellow View Post
    I want to find prime and jacobson radical radicals in Z[T]/(T^3), here Z = integers.
    My definition of jacobson radical is that it is intersection of maximal ideals and prime radical is intersection of prime ideals.
    Please help! Thanks
    let R=\frac{\mathbb{Z}[T]}{<T^3>}. recall that N(R), prime radical, is also the set of all nilpotent elements of R. now it should be obvious to you that N(R)=\frac{<T>}{<T^3>}.

    to find J(R), Jacobson radical, we use this fact that in a (commutative) ring S, a \in J(S), if and only if 1-ax is a unit for all x \in S. it's very easy to

    see that the units of R are \pm 1 + \alpha T + \beta T^2 + <T^3>, \ \ \alpha, \beta \in \mathbb{Z}. from here conclude that J(R)=\frac{<T>}{<T^3>}. \ \ \ \square
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  3. #3
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    you are saying that the set of prime radical is same as set of nilpotents, fine but How do you get that the prime radical is $<T>/<T^3>$.

    and how do you know that $\pm 1 + \alphaT + \betaT + <T^3>$ is a unit in R but if this is a unit how can you conclude that jacobson radical is $<T>/<T^3>$.
    please help thnaks.
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  4. #4
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    Quote Originally Posted by peteryellow View Post
    you are saying that the set of prime radical is same as set of nilpotents, fine but How do you get that the prime radical is $<T>/<T^3>$.
    an element of R is in the form u=p(T) + <T^3>, where p(T) \in \mathbb{Z}[T]. ( by the way we may assume that p(T) is of degree at most 2 because

    the terms of degree 3 or bigger belong to <T^3>. ) now u^n = 0 means (p(T))^n \in <T^3>, which is possible for some n if and only if the constant

    term of p(T) is 0. thus u is nilpotent if and only if p(T) \in <T>.

    the second part of your question (Jacobson radical) is as easy as this part. just follow the line of the proof that i gave you.
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