### 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 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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