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Math Help - Proving L is an Ideal - Part of the Proof of Hilbert's Bass Theorem

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    Super Member Bernhard's Avatar
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    Proving L is an Ideal - Part of the Proof of Hilbert's Bass Theorem

    I am reading Dummit ad Foote's proof of Hilbert's Basis Theorem (See attached for the theorem and proof)

    In the proof I is an ideal in R[x] L is the set of all leading coefficients of elements of I

    D&F then proceed to prove that L is an ideal of R

    Basically they establish that if elements a and b belong to L and r belongs to L then ra - b belongs to L.

    D&F claim that this shows that L is an ideal but for an ideal we need to show that for  a, b \in L and  r \in R we have:

     a - b  \in L and  ra \in R

    My question is how exactly does  ra - b \in L  \Longrightarrow a - b  \in L and  ra \in R ??

    Peter
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    Re: Proving L is an Ideal - Part of the Proof of Hilbert's Bass Theorem

    Quote Originally Posted by Bernhard View Post

    My question is how exactly does  ra - b \in L  \Longrightarrow a - b  \in L and  ra \in R ??
    Take  r=1 in the first case and b= 0 in the second.
    Thanks from Bernhard
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