
Noncommutative Associative Algebras 27
Example 2.3.2.7. Let X = {a, b}, and let s
1
= a
2
− 1, s
2
= ab − a, and
f = a
2
b. Then r
s
1
(f) = b, and r
s
2
(f) = a
2
. Therefore, if we order S = {s
1
, s
2
}
in a way that s
1
< s
2
, then the long division terminates after just one reduc-
tion, and returns b. If we order S = {s
1
, s
2
} in a way that s
1
> s
2
, then the
long division takes two reductions, and returns 1.
Lemma 2.3.2.8. For every f ∈ T (X), the long division algorithm terminates
in a finite number of steps. Its output is an element
˜
f reduced with respect to
S, for which lt(
˜
f) lt(f) and f + (S) =
˜
f + (S).
Proof. By Lemma 2.3.2.5, the leading monomial of the dividend ...