A technical point, which otherwise I forget:

In Lemma 2.9 (Buchberger's Fisrst Criterion) there is $Q$ which should be ${\Cal Q}$. Please correect it
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>> Remove Theorem 2.11; it is trivial and completely useless.
>> Please moreover note (in page 9) that Theorem 2.11 allows to TEST whether a basis is Groebner but is NOT SUFFICENT TO COMPUTE IT.
>
> Th 2.11 can be used to compute a G-basis, 

Not exactly. Once you have a basis you test whether it is Groebner using Th.2.12 and, in case not all NormalForm are zero you have a negatve answer and must:
*) make a list of further S-pairs,
*) apply to it Gebauer-Moeller result
*) apply Gebauer-Moeller also to the corrent list of S-pairs  to remove pairs which now becme useless
*) and then you don't exactly apply Th. 2.11; because you apply that test only to a suitable subset (not the whose set) of S-pairs.

The point is that (as it is explicitly stated at the end of 2.3.) the algorithm presented there (and not the one implied by Th.2.11) is "how close we are to the finite field case"

The algorithm implied by Section 2.3. extends verbatim the proper best avaialble Buchberger Algorithm which is the one describe in my introductory section of the book (see Figure 5 in the enclosed file).

The formulation of Section 2.3. has been carefully done to present a Buchberger Algorithm over a PIR which is completely indentical with the  field case.

I agree with you that the formulation around Th.2.11 is not well formulated beacuse it does not speak of normal form.

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Given any $\theta \in R$, there exist
$u \in R^*$, and a unique non-negative integer $\nu(\theta)$ such that $\theta = u \pi^{\nu(\theta)}$.
With respect to this notation $\theta$ has nilpotency $n-\nu(\theta)$.

The algorithm of Figure 2 combined with M\"oller's result has the following form for $R$.
We use the same notation as Section~\ref{Moller}.

\begin{theorem}
\label{thbuch3}Let $G=\{g_{i}\}_{1}^{s}$ be a set of
non-zero polynomials in $R[\mathbf{x}]$ with
$\emph{\limfunc{{\bf M}}}(g_{i})=\theta _{i}\pi^{t_{i}}\tau_{i},i=1,\ldots
,s.$ Then $G$ is a Gr\"{o}bner basis if and only if
$NF(B(i,j), G ) = 0$ and
$NF(\pi^{n-t_{i}}g_{i}, G)= 0$,
for all $i,j\in \{1...s\}$ with $t_i \leq t_j$.
\end{theorem}

For computational purposes, we will assume that a Gr\"{o}bner basis $G=\{g_1,...,g_s\}$ over $R$ is {\em minimal}, so that ${\bf M}(g_i)$ does not divide ${\bf M}(g_j)$ for $i \neq j$ and that $\lc(g_i) =  c_i = \pi^{\ell_i}, 1 \leq \ell_i = \nu(\lc(g_i)) < n$ (for the finite field case, this reduces to the assumption that $\lc(g_i)=1$ for each $i$).

M\"oller's adaptation of Buchberger's Algorithm in the case of finite chain ring
requires, in each loop, to compute
\begin{itemize}
\item the (useful!) S-pairs
$B(i,j) := \pi^{t_j-t_i} \frac{\lcm(\tau_i,\tau_j)}{\tau_i} g_i -\frac{\lcm(\tau_i,\tau_j)}{\tau_j}g_j$
with $t_i \leq t_j$ and
\item the annihilator-pairs $\pi^{t_i}g_i$.
\end{itemize}
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Given any $\theta \in R$, there exist
$u \in R^*$, and a unique non-negative integer $\nu(\theta)$ such that $\theta = u \pi^{\nu(\theta)}$.
With respect to this notation $\theta$ has nilpotency $n-\nu(\theta)$.

For computational purposes, we will assume that a Gr\"{o}bner basis $G=\{g_1,...,g_s\}$ over $R$ is {\em minimal}, so that ${\bf M}(g_i)$ does not divide ${\bf M}(g_j)$ for $i \neq j$ and that $\lc(g_i) =  c_i = \pi^{\ell_i}, 1 \leq \ell_i = \nu(\lc(g_i)) < n$ (for the finite field case, this reduces to the assumption that $\lc(g_i)=1$ for each $i$).

M\"oller's result gives for $R$ {\em verbatim} the version of Buchberger Algorithm described in \cite{Teo} Figure 5.


In the case of finite chain ring
it consists, in each loop, to compute normal forms of
\begin{itemize}
\item the (useful!) S-pairs
$B(i,j) := \pi^{t_j-t_i} \frac{\lcm(\tau_i,\tau_j)}{\tau_i} g_i -\frac{\lcm(\tau_i,\tau_j)}{\tau_j}g_j$
with $t_i \leq t_j$ and
\item the annihilator-pairs $\pi^{t_i}g_i$.
\end{itemize}

It implies the following test for a basis to be Groebner:

\begin{theorem}
\label{thbuch3}Let $G=\{g_{i}\}_{1}^{s}$ be a set of
non-zero polynomials in $R[\mathbf{x}]$ with
$\emph{\limfunc{{\bf M}}}(g_{i})=\theta _{i}\pi^{t_{i}}\tau_{i},i=1,\ldots
,s.$ Then $G$ is a Gr\"{o}bner basis if and only if
$NF(B(i,j), G ) = 0$ and
$NF(\pi^{n-t_{i}}g_{i}, G)= 0$,
for all $i,j\in \{1...s\}$ with $t_i \leq t_j$.
\end{theorem}
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