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

I have a second problem (see previous post for first problem) understanding a step in the proof of Proposition 24, Page 322 of D&F

Proposition 24 reads as follows:

Proposition 24.Fix a monomial ordering on and let I be a non-zero ideal in R

(1) If are any elements of I such that

then is a Grobner Basis for I

(2) The ideal I has a Grobner Basis

The proof of Proposition 24 begins as follows:

Proof:Suppose with .

We need to see that generate the ideal I.

If use general polynomial division to write where no non-zero term in the remainder r is divisible by any

Since , also , which means LT9r) is in LT(I).

But then LT(r) would be divisible by one of , which is a contradiction unless r= 0... ... etc etc

... ... ...

My problem is with the last (bold) statement - as follows:

We have that

Now since , then we have that r is a finite sum of the form

... ... ... (*)

where and

But surely (*) is NOT guaranteed to be divisible by for some i

Can someone please clarify this issue?

Peter