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Nilradical math help

  1. Feb 6, 2008 #1
    1. The problem statement, all variables and given/known data
    Let N be an ideal in of a commutative ring R. What is the relationship of the ideal [itex]\sqrt{N} [/itex] to the nilradical of R/N? Word your answer carefully.

    Recall that the nilradical of an ideal N is the collection of all elements a in R such that a^n is in N for some n in Z^+.

    EDIT: this definition is dead wrong

    2. Relevant equations



    3. The attempt at a solution
    Answer: a is in the nilradical of N iff (a+N) is in the nilradical of R/N. So they are the same. Why did they say word your answer carefully?
     
    Last edited: Feb 7, 2008
  2. jcsd
  3. Feb 6, 2008 #2

    Hurkyl

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    Are you quite sure that's the definition of "nilradical" given in your class? The definition you give is indeed the definition of [itex]\sqrt{N}[/itex] but that's called the "radical of N". The term "nilradical" applies to rings, and the nilradical of R is the radical of its zero ideal. (i.e. the set of nilpotent elements of R)

    They're obviously not "the same"; one is an ideal of R, the other is an ideal of R/N, and they (usually) don't have a single element in common.
     
    Last edited: Feb 6, 2008
  4. Feb 7, 2008 #3
    Sorry. I was totally wrong.

    The radical of an ideal N is defined as the set [itex] \sqrt{N} [/itex] of all a in R such that a^n is in N for some positive integer n.

    The nilradical of a ring is the collection of all the nilpotent elements.

    Let me see if I can figure it out with the correct definitions...
     
  5. Feb 7, 2008 #4
    Is this answer worded correctly:

    a is in the radical of N iff (a+N) is in the nilradical of R/N

    So they are canonically homomorphic, right? I would call them the same but it seems like other people disagree...
     
  6. Feb 7, 2008 #5

    morphism

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    Looks good.

    "Canonically homomorphic"?
     
  7. Feb 7, 2008 #6
    OK. Forget that.

    My final answer is "a is in the radical of N iff (a+N) is in the nilradical of R/N".

    Why did they say word your answer carefully? Is my answer worded carefully enough?
     
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