I am reading Dummit and Foote Section 9.6 Polynomials in Several Variables Over a Field and Grobner Bases.
I have a 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 ... ... etc etc
My question: How do we know ?
Would appreciate some help