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Math Help - Ideals

  1. #1
    MHF Contributor chiph588@'s Avatar
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    Ideals

    I'm new to this topic, so any help would be appriciated!

    Note: these are all in a communitive ring  R .

    1.  a \mid b \Longleftrightarrow (b) \subseteq (a) .

    2.  u \in R is a unit  \Longleftrightarrow (u) = R .

    3.  a and  b are associative  \Longleftrightarrow (a) = (b) .

    4.  p is prime  \Longleftrightarrow ab \in (p) implies that either  a \in (p) or  b \in (p) .
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  2. #2
    Senior Member vincisonfire's Avatar
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    1.  (a) = \{ ra : r \in R \}
    If  a \mid b then   b =qa for some q \in R Do you see why  (b) \subseteq (a)<br />
?
    2. If  u \in R<br />
is a unit then there exist  u^{-1} \in R such that  uu^{-1} = 1 .
     (u) = \{ ru : r \in R \} If   r = qu^{-1} ,  q \in R  then we see that it is all the multiple of 1 and thus the ring itself.
    3. I don't know what associative is sorry.
    4.  (p) = \{ rp : r \in R \}
     ab \in (p) \implies  ab = rp  for some  r \in R . Since p is prime, either a or b has to have p in its prime factorization. It follows that <br /> <br />
a \in (p)<br />
or <br /> <br />
b \in (p)<br />
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    for #1.

     (a) = \{ ra : r \in R \}

    Since  b = qa , \; (b) = \{ (rq)a : r \in R \}

    So my question is do we know  rq \in R ?
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  4. #4
    Senior Member vincisonfire's Avatar
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     (a) = \{ ra : r \in R \} This is all the multiples of a.
     a \mid b So b is a multiple of a.
     (b) = \{ rb : r \in R \} Multiples of b or multiples of multiples of a. So yes we know that  rq \in R . You could also simply use the fact that a ring is closed under multiplication.
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    ah yes, I forgot about closure!
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