
Noncommutative Associative Algebras 37
Proposition 2.4.2.2. If Algorithm 2.4.2.1 terminates then its output is the
reduced Gröbner basis of I.
Proof. Immediate corollary to Theorem 2.4.1.5.
In many computations, instead of explicitly finding the standard form at
each step, we will merely underline the leading monomial (which is being
reduced).
Example 2.4.2.3. For the quantum plane from Example 2.1.1.3, the noncom-
mutative Buchberger algorithm terminates instantly, since the only leading
monomial (for glex order with x
1
≺ x
2
) is x
2
x
1
which has no self-overlaps.
Example 2.4.2.4. We consider the ideal (y
2
+ x
2
) of the tensor algebra
T (x, y), and impose the ...