| ### 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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