I am reading Dummit and Foote Section 9.6 Polynomials In Several Variables Over a Field and Grobner Bases

I have a very basic question regarding the beginning of the proof of Hilbert's Basis Theorem (see attachment for a statement of the Theorem and details of the proof)

Theorem 21 (Hilbert's Basis Theorem) If R is a Noetherian ring then so is the polynomial ring R[x]

The proof begins as follows:

Proof: Let I be an ideal in R[x] and let L be the set of all leading coefficients of the elements in I. We will first show that L is an ideal of R, as follows. Since I contains the zero polynomial, $\displaystyle 0 \in L $.

Let $\displaystyle f = ax^d + ... $ and $\displaystyle g = bx^e + ... $ be polynomials in I of degrees d, e and leading coefficients $\displaystyle a, b \in R $.

Then for any $\displaystyle r \in R $ either ra - b is zero or it is the leading coefficient of the polynomial $\displaystyle rx^ef - x^dg $. Since the latter polynomial is in I ... ...???

How do we know that the polynomial $\displaystyle rx^ef - x^dg $ is in I?My problem:

For $\displaystyle rx^ef $ to belong to I we need $\displaystyle rx^e \in I $. Now it seems to me that $\displaystyle rx^e \in I $ if $\displaystyle x^e \in I $ (right?) but how do we know that or be sure that $\displaystyle x^e \in I $?

Can someone clarify this situation for me?

Peter