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 and we have:
My question is how exactly does and ??