| ### 7.9.3 Groebner bases for two-sided ideals in free associative algebras We say that a monomial divides (two-sided or bilaterally) a monomial , if there exist monomials , such that , in other words is a subword of . For a subset , define the 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 . Suppose, that the weights of the ring variables are strictly positive. We can interprete these weights as defining a nonstandard grading on the ring. If the set of input polynomials is weighted homogeneous with respect to the given weights of the ring variables, then computing up to a weighted degree (and thus, also length) bound results in the truncated Groebner basis . In other words, by trimming elements of degree exceeding from the complete Groebner basis , one obtains precisely . In general, given a set , which is the result of Groebner basis computation up to weighted degree bound , then it is the complete finite Groebner basis, if and only if holds.

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