Let $I$ be an ideal of a polynomial ring $R$. Fix a monomial order. Denote the $S$-polynomial of $f, g\in R$ by $S(f, g)$ and denote the gcd of their leading terms by $T(f, g)$.
Consider the following variation of the Buchberger algorithm
- Input: A set of polynomials $F$ that generates the ideal $I$
Output: A Gröbner basis $G$ for $I$
- $G :=F$
- For every $f_i$, $f_j$ in $G$, if $T(f_i, f_j)\neq 1$, add $S(f_i, f_j)$ to $G$.
- Repeat 2. until for every $f_i, f_j$ in $G$ we have $T(f_i, f_j)=1$ or $S(f_i, f_j)\in G$.
It can be shown (using a variant of Buchberger criterion) that if this algorithm terminates then $G$ is a Gröbner basis for $I$.
My question is: is it true that the algorithm always terminates?