| ### 7.9.4 Bimodules and syzygies and lifts Let 411#411 300#300,..., 301#301 be the free algebra. A free bimodule of rank 299#299 over 190#190 is 412#412,where 413#413 are the generators of the free bimodule. NOTE: these 413#413 are freely non-commutative with respect to elements of 190#190 except constants from the ground field 50#50. The free bimodule of rank 1 414#414 surjects onto the algebra 190#190 itself. A two-sided ideal of the algebra 190#190 can be converted to a subbimodule of 414#414. The syzygy bimodule or even module of bisyzygies of the given finitely generated subbimodule 415#415is the kernel of the natural homomorphism of 190#190-bimodules 416#416that is 417#417 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 418#418 of a bimodule 13#13, the lift(ing) process returns a matrix, which encodes the expression of generators 419#419 in terms of generators of 420#420 like this: 421#421 where 422#422 are elements from the enveloping algebra 423#423encoded as elements of the free bimodule of rank 295#295, namely by using the non-commutative generators of the free bimodule which we call `ncgen`.

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