
Noncommutative Associative Algebras 57
Computing the reduced form of g
0
2
with respect to {g
1
}, we obtain:
(baba −ab
2
a) −b(aba −ba) = −ab
2
a + b
2
a −→ ab
2
a −b
2
a = g
2
.
We obtain a new bigger self-reduced set:
R
1
= { g
1
= aba −ba, g
2
= ab
2
a −b
2
a }.
Again, we see that lm(g
2
) = ab
2
a for any monomial order. The second itera-
tion produces three S-polynomials:
g
1
b
2
a −abg
2
= (aba −ba)b
2
a −ab(ab
2
a −b
2
a) −→ bab
2
a −ab
3
a = g
0
3
,
g
2
ba −ab
2
g
1
= (ab
2
a −b
2
a)ba −ab
2
(aba −ba) −→ b
2
aba −ab
3
a = g
0
4
,
g
2
b
2
a −ab
2
g
2
= (ab
2
a −b
2
a)b
2
a −ab
2
(ab
2
a −b
2
a) −→ b
2
ab
2
a −ab
4
a = g
0
5
.
Computing the normal forms of g
0
3
, g
0
4
, g
0
5
with respect to {g
1
, g
2
} gives:
(bab
2
a −ab
3
a) −b(ab
2
a −b
2
a) = −ab
3
a + b