An elimination lemma for algebras with PBW bases

Let K be a field, and A=K[a1,…,an] a finitely generated K-algebra with the PBW K-basis ℬ={a1α1⋯anαn|(α1,…,αn)∈ℕn}. It is shown that if L is a nonzero left ideal of A with GK.dim(A∕L) = d<n ( = the number of generators of A), then L has the elimination property in the sense that V(U)∩L≠{0} for every subset U={ai1,…,aid+1}⊂{a1,…,an} with i1<i2<⋯<id+1, where V(U) = K-span{ai1α1⋯aid+1αd+1|(α1,…,αd+1)∈ℕd+1}. In terms of the structural properties of A, it is also explored when the condition GK.dim(A∕L)<n may hold for a left ideal L of A. Moreover, from the viewpoint of realizing the elimination property by means of Gröbner bases, it is demonstrated that if A is in the class of binomial skew polynomial rings or in the class of solvable polynomial algebras, then every nonzero left ideal L of A satisfies GK.dim(A∕L)< GK.dimA = n ( = the number of generators of A), thereby L has the elimination property.