On the Gr\"obner bases in ore extensions
Ìóøðóá Â.À.
Gr\"obner (standard) bases of ideals of skew polynomial rings have
been of interest in the last few years.
In 1992 V.~Weispfenning introduced and studied
the construction of Gr\"obner bases for
one-sided and two-sided ideals of skew polynomial rings
with one substitution operator over a polynomial ring
in one variable.
This line of investigation was continued by M.~Pesch.
Let $K$ be a field, $N\ge 2$, $e_1$,
$e_2$,\ldots,$e_N$ some positive integers,
$\lambda_1,\ldots,\lambda_N $ some non-zero elements of
$K$, and $ \sigma $ the cycle of the length $N$
such that $\sigma(i)=i+1$ for $1\le i\le N-1$ and
$\sigma(N)=1$.Let $R=K[x_1,x_2,\ldots, x_N]$.
By $\phi $ we denote an injective ring endomorphism of $K$-algebra
$R$,such that $\phi(x_i)= \lambda_i x_{\sigma(i)}^{e_i}+f_i, \;
\mbox{ where } f_i\prec x_i^{e_i}.$
Consider the skew polynomial ring (Ore extension) $R[y,\phi,\delta]$
over $R$ as a $K$-algebra. By the terms of this algebra
we mean the products of the following form
${x_1^{k_1}\dots x_N^{k_N}y^m}$,
where $k_1,\ldots, k_N, m$ are non-negative integers.
We associate to $R[y,\phi,\delta]$ the monoid of
terms $F_{\alpha}[y]$, multiplication in which is given by the
relations $yx_i=x_{\sigma(i)}^{e_i}y$.
By {\sl a term order} we mean an
admissible well-ordering on the monoid $F_{\alpha}[y]$.
Let $e=e_1e_2\cdot\cdot\cdot e_N$.
Set
$$\theta(x_1^{k_1}\dots x_N^{k_N})=k_1e+ \sum_{i=2}^N k_i
\frac{e^{1+\frac{i-1}{N}}}{e_1\cdot\cdot\cdot e_{i-1}}.
$$
\begin{Theorem} \label{END}
If the polynomial $ x^N-e $ is irreducible over $
\mathbb Q $, then the monoid of terms $F_{\alpha}[y]$
possesses a unique term order.
If this is the case then
the only term order $\prec $ is given by the rule:
$$
x^ay^n \prec x^b y^m \Leftrightarrow (n,\theta(x^a)) < (m,
\theta(x^b)),
$$
where $< $ is the lexicographical order on the set of
pairs of real numbers.
Furthermore, if the polynomial $x^N-e$ is reducible over
$\mathbb Q$,
then there are no term orders.
\end{Theorem}
Suppose now that
the polynomial $x^N-e$ will always be
irreducible over the field of rational numbers.
Let $L$ be a left (right) ideal of $R[y, \phi, \delta]$,
and $B$ a subset of $L$. Then $B$ is called a Gr\"obner
(standard) basis of $L$ if,
for each nonzero element $f$ of $L$, there exists an
element $p\in B$ such that the highest term of
$p$ divides the highest term of
$f$ on the right in the monoid of terms.
\begin{Theorem}\label{main}
There exists an algorithm that computes by a finite
number of steps a Gr\"obner bases of the left (right) ideal
generated by $B$, starting from a given finite subset
$B$ of $R[y,\phi,\delta]$.
\end{Theorem}
The class of algebras considered in this note
exibits the following phenomenon in Gr\"obner basis
theory. These algebras
need not be left Noetherian and need not be right Noetherian. Nevertheless
every finitely generated ideal of these algebras has a
finite Gr\"obner basis.