  Let $K$ be an infinite field and $K< X> =K< X_1,...,X_n>$ the free
associative algebra generated by $X=\{X_1,...,X_n\}$ over $K$. It is proved
that if $I$ is a two-sided ideal of $K< X>$ such that the $K$-algebra $A=K< X>
/I$ is almost commutative in the sense of [3], namely, with respect to its
standard $\mathbb{N}$-filtration $FA$, the associated $\mathbb{N}$-graded
algebra $G(A)$ is commutative, then $I$ is generated by a finite Gr\"obner
basis. Therefor, every quotient algebra of the enveloping algebra
$U(\mathbf{g})$ of a finite dimensional $K$-Lie algebra $\mathbf{g}$ is, as a
noncommutative algebra of the form $A=K< X> /I$, defined by a finite Gr\"obner
basis in $K< X>$.

Rings and Algebras