An elimination lemma for algebras with PBW bases

Abstract

ABSTRACTLet K be a field, and a finitely generated K-algebra with the PBW K-basis . 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 with , where V(U) = K-span. 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.