### 7.6.2 Groebner bases for two-sided ideals in free associative algebras

We call a total ordering on the free monoid (where is identified with the identity element) a monomial ordering if the following conditions hold:

- is a well-ordering on , that is ,
- , if , then ,
- , if and , then .

Hence the notions like a leading monomial and a leading coefficient transfer to this situation.

We say that a monomial divides monomial , if there exist monomials , such that .

In other words is a proper subword of .

For a subset , define a leading ideal of to be the two-sided ideal .

Let be a fixed monomial ordering on . We say that a subset is a (two-sided) Groebner basis for the ideal with respect to , if . That is there exists , such that divides .

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