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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
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 [TEX] a, b \in L [/TEX] and [TEX] r \in R[/TEX] we have:

[TEX] a - b \in L [/TEX] and [TEX] ra \in R [/TEX]

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

Peter


[This has also been posted on MHF]
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,705
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 [TEX] a, b \in L [/TEX] and [TEX] r \in R[/TEX] we have:

[TEX] a - b \in L [/TEX] and [TEX] ra \in R [/TEX]

My question is how exactly does [TEX] ra - b \in L \Longrightarrow a - b \in L [/TEX] and [TEX] ra \in R [/TEX]??
If you know that $ra-b\in L$ whenever $a,b\in L$ and $r\in R$ then in particular this will hold when $b=0$, so that $ra\in L$; and also when $r=1$ (the identity element of $R$) so that $a-b\in L$.