|  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  . 
The notion of Groebner-Shirshov basis applies to more general algebraic structures,
but means the same as Groebner basis for associative algebras.
 
Suppose, that the weights of the ring variables are strictly positive.
We can interprete these weights as defining a non-standard 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. 
Note: If the set of input polynomials is not weighted homogeneous with respect to the 
 weights of the ring variables, and a Groebner is not finite, then actually not much can be
 said precisely on the properties of the given ideal. By increasing the length bound bigger generating
 sets will be computed, but in contrast to the weighted homogeneous case some polynomials in
 of small length first enter the basis after computing up to a much higher length bound.
 
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