  Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a finitely generated
$K$-algebra with the PBW $K$-basis ${\cal B}=\{a_{1}^{\alpha_1}\cdots
a_{n}^{\alpha_n}~|~(\alpha_1,\ldots ,\alpha_n)\in\mathbb{N}^n\}$. 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 {\it elimination property} in
the sense that ${\bf V}(U)\cap L\ne \{0\}$ for every subset $U=\{
a_{i_1},\ldots ,a_{i_{d+1}}\}\subset\{a_1,\ldots ,a_n\}$ with $i_1<i_2<\cdots
<i_{d+1}$, where ${\bf V}(U)=K$-span$\{a_{i_1}^{\alpha_1}\cdots
a_{i_{d+1}}^{\alpha_{d+1}}~|~(\alpha_1,\ldots
,\alpha_{d+1})\in\mathbb{N}^{d+1}\}$. 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 Gr\"obner bases, it is demonstrated that if $A$ is in the
class of binomial skew polynomial rings [G-I2, Serdica Math. J., 30(2004)] or
in the class of solvable polynomial algebras [K-RW, J. Symbolic Comput.,
9(1990)], then every nonzero left ideal $L$ of $A$ satisfies GK.dim$(A/L)<$
GK.dim$A=n$ ($=$ the number of generators of $A$), thereby $L$ has the
elimination property.
