|  7.9.4 Bimodules and syzygies and lifts 
Let 
   ,...,  be the free algebra.
A free bimodule of rank  over  is  ,where  are the generators of the free bimodule. 
NOTE: these 
 are freely non-commutative with respect to
elements of  except constants from the ground field  . 
The free bimodule of rank 1 
 surjects onto the algebra  itself.
A two-sided ideal of the algebra  can be converted to a subbimodule of  . 
The syzygy bimodule or even module of bisyzygies
of the given finitely generated subbimodule
 is the kernel of the natural homomorphism of  -bimodules  that is  
The syzygy bimodule is in general not finitely generated. 
Therefore as a bimodule, both the set of generators of the 
syzygy bimodule and its Groebner basis 
are computed up to a specified length bound.
 
Given a subbimodule 
 of a bimodule  , the lift(ing) process
returns a matrix, which encodes the expression of generators  in terms of generators of  like this:  
where T_ij are elements from the enveloping algebra 
 encoded as elements of the free bimodule of rank  ,
namely by using the non-commutative generators of the
free bimodule which we call ncgen. 
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