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