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, .
Let and be polynomials in I of degrees d, e and leading coefficients .
Then for any either ra - b is zero or it is the leading coefficient of the polynomial . Since the latter polynomial is in I ... ...???
My problem: How do we know that the polynomial is in I?
For to belong to I we need . Now it seems to me that if (right?) but how do we know that or be sure that ?
Can someone clarify this situation for me?