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Prime and jacoboson radical

  1. Aug 27, 2008 #1
    I want to find prime and jacobson radical radicals in Z[T]/(T^3), here Z = integers.

    Is it true that Z[T]/(T^3) is a field, because T^3 is irreducibel over Z[T]. If it is true that
    Z[T]/(T^3) is a field then 0 is the prime and jacobson radical radical.

    Is it true please help.
     
  2. jcsd
  3. Aug 27, 2008 #2
    No T^3 is not irreducible, so can some body help me that how does prime and maximal ideals in Z[T]/(T^3) look like.
     
  4. Aug 27, 2008 #3

    morphism

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    Think about this abstractly: if R is a ring with an ideal I, what can you say about the ideal structure of R/I?
     
  5. Aug 27, 2008 #4

    mathwonk

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    if x is nilpotent, i.e. some positive power of x is zero, then what prime ideals does x lie in?


    what about the converse question? if x is not nilpotent, can you find a prime ideal not containing x? (do you know about localizing a ring at powers of an element?)
     
  6. Aug 28, 2008 #5
    No I dont understand what you are saying mathwonk. My definition of jacobson radical is that it is intersection of maximal ideals and prime radical iks intersection of prime radical.
     
  7. Aug 28, 2008 #6

    morphism

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    mathwonk is alluding to the fact that the prime radical of (a commutative ring) R is nothing but the set of all nilpotent elements in R. The same comment applies to the Jacobson radical of R whenever R is finitely-generated (as a Z-module).
     
  8. Aug 28, 2008 #7
    But Morphism can you please tell me that what are prime and maximal ideals of the ring, and how can I FIND THEM. PLEASE HELP. THNAKS
     
  9. Aug 28, 2008 #8

    morphism

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    But why? mathwonk's method is much more elegant.
     
  10. Aug 28, 2008 #9

    mathwonk

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    the point is that if you are looking for prime ideals you should look at nilpotent elements.
     
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