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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 $\displaystyle a, b \in L $ and $\displaystyle r \in R$ we have:

$\displaystyle a - b \in L $ and $\displaystyle ra \in R $

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

Peter

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

Quote:

Originally Posted by

**Bernhard**

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

Take $\displaystyle r=1$ in the first case and $\displaystyle b= 0$ in the second.