When generalizing Gr\"obner bases to other algebras, in a first account we have to consider
 rings as coefficient domains.
In Section \ref{section.buchberger} we have seen that 
 already in $\z[X_1, \ldots, X_n]$ the existence of a standard representation
 no longer implies reducibility and has to be modified in order to do this.
In the setting of function rings such a modification could be follows:
\begin{definition}\label{def.standard.rep.mod}~\\
{\rm
Let $F$ be a set of polynomials in $\f$
 and $g$ a non-zero polynomial in $\ideal{r}{}(F)\subseteq \f$.
A representation of the form
 \begin{eqnarray}
  g & = & \sum_{i=1}^n f_i \rmult m_i, 
   f_i \in F, m_i \in \monoms(\f), n \in \n
 \end{eqnarray}
 where additionally $\hterm(g) = \hterm(f_1 \rmult m_1)$ and
 $\hterm(g) \succ \hterm(f_i \rmult m_i)$ holds for
 $2 \leq i \leq n$ is called a
 \betonen{strong right standard representation} of $g$ in terms of $F$.
\dend
}
\end{definition}
One possible reduction relation related to such strong standard representations
 is called \betonen{strong reduction}\label{page.strong.reduction}\footnote{Strong reduction has been studied extensively
 for monoid rings in \cite{Re95}.} where a monomial $m_1$ is reducible
 by some polynomial $f$ if there exists some monomial $m_2$ such that
 $m_1 = \hm(f \rmult m_2)$.
Notice that such a reduction step eliminates the occurence of the term $\hterm(m_1)$
 in the resulting reductum.
When generalizing this reduction relation to function rings we can no longer
 localize the reduction step to checking whether $\hm(f)$ divides $m_1$, as
 now the whole polynomial is involved in the reduction step.
We can no longer conclude that $\hm(f)$ divides $m_1$ but only that
 $m_1 = \hm(f \rmult m_2)$.
We cannot even conclude that some term in $\terms(f)$ divides $\hterm(m_1)$ as
 while $\hterm(f \rmult m_2)$ might result from some $t \in \terms(f)$ still
 $\hterm(t \rmult m_2) \neq \hterm(f \rmult m_2)$ is possible.
Let us look at the skew-polynomial ring over $\q$ and
 $X_1 \succ X_2 \succ X_3 \succ X_4$ with
 $X_2 \rmult X_1 = X_3$, $X_3 \rmult X_1 = -X_3 + X_4$, $X_3 \rmult X_2 = X_2X_3$,
 and $X_i \rmult X_j = X_iX_j$, $i \leq j$.
Then for $f = X_2 + X_3$ and $m_2 = X_1$ we get
 $f \rmult m_2 = (X_2 + X_3) \rmult X_1 = X_3 - X_3 + X_4 = X_4$.
Hence $X_4$ results from $X_3 \rmult X_1$ but
 $X_4 \neq \hterm(X_3 \rmult X_1) = X_3$.

An equivalent algebraic definition of a Gr\"obner basis now has to describe
 the set $\hm(\ideal{}{}(G))$ in terms of the polynomials in $G$.
The simple analogon for right ideals in function rings
 $\hm(\ideal{r}{}(G)) = \{  m_1 \rmult m_2  \mid
 m_1 \in \hm(G), m_2 \in \monoms(\f) \}$ will only work if $\rr$ is a reduction ring,
 $\rmult: \myt \times \myt \myr \myt$, and
 the ordering on $\myt$ is compatible with multiplication from the right. 
Notice that such a characterization can at most describe a weak right Gr\"obner basis
 as defined in Definition \ref{def.weak.gb}, as a reduction relation based on divisibility of
 monomials without remainders can at most achieve that any element of the right ideal
 reduces to zero.
If we have $\rmult: \myt \times \myt \myr \f$ instead, a refinement to
 $\hm(\ideal{r}{}(G)) = \{  \hm(m_1 \rmult m_2)  \mid
 m_1 \in \hm(G), m_2 \in \monoms(\f) \} = \hm(\{  m_1 \rmult m_2  \mid
 m_1 \in \hm(G), m_2 \in \monoms(\f) \})$ might seem possible.
But the main problem is that in many cases 
 orderings on $\myt$ are not compatible with the multiplication
 and such a definition requires that the head monomial of a generating polynomial
 for the right ideal essentially describes its influence on the head monomials
 of the right ideal elements.
However, in our very general setting this is no longer so.

Let us recall Example \ref{exa.not.stable}
  where the ordering $\succeq$ induced by $x \succ 1$ on terms respectively monomials
 is well-founded but in general not
 compatible with multiplication, due to the
 algebraic structure of $\myt$.
There for the polynomial $f = x + 1$ and the term $x$ we get
 $\hm(f \rmult x) = x$ while $\hm(f) \rmult x = 1$.\label{page.not.stable}

Hence, in the general case where we have $\rmult: \myt \times \myt \myr \f$
 a characterization has to include the whole polynomial, namely
 $\hm(\ideal{r}{}(G)) = \{  \hm(f \rmult m)  \mid
 f \in G, m\in \monoms(\f) \}$.  

Behind these phenomenon lies the fact that the  definition
 of divisors arising from this algebraic characterization of Gr\"obner bases
 in the context of function rings does not have the same
 properties as divisors in polynomial rings.
Now $m_1 \in \hm(\ideal{r}{}(G))$ implies the existence of $m_2 \in \monoms(\f)$
 such that $\hm(f \rmult m_2) = m$.
This definition of right divisors is no longer transitive.
Reviewing the previous example we see that for $f = x + 1$, $m_2 = x$ and $m = \hm(f) = x$ we get $\hm(f \rmult m_2) = \hm((x+1) \rmult x) = x$, i.e.~$\hm(f\rmult m_2)$
 divides $m$.
On the other hand $x$ divides $1$ as $x \rmult x = 1$.
But $\hm(\hm(f \rmult m_2) \rmult x) = 1$ while
 $\hm(f \rmult m_2 \rmult x) = x$.

Notice that even if we locailze the concept of right divisors to monomials we do not
 get transitivity.
We are interested when for some monomials $m_1,m_2,m_3 \in \monoms(\f)$
 the facts that $m_1$ divides $m_2$ and $m_2$ divides $m_3$ implies that
 also $m_1$ divides $m_3$.
Let $m, m` \in \monoms(\f)$ such that
 $\hm(m_1 \rmult m) = m_2$ and $\hm(m_2 \rmult m`) = m_3$.
Then $m_3 = \hm(m_2 \rmult m`) = \hm( \hm(m_1 \rmult m) \rmult m`)$.
When does this equal $\hm(m_1 \rmult m \rmult m`)$?\label{page.divisor}
Obviously if we have $\rmult : \monoms(\f) \times \monoms(\f) \mapsto \monoms(\f)$
 this is true.
However if multiplication of monomials results in polynomials we are in trouble.
Let us look at the skew-polynomial ring $\q[X_1,X_2,X_3]$ associated
 to the set of terms defines in Example \ref{exa.skew}, i.e.~$X_2 \rmult X_1 =
 X_2 + X_3$ and $X_3 \rmult X_1 = X_1$.
Then from the fact that $X_2$ divides $X_2$ we get
 $\hm(X_2 \rmult X_1) = X_2$ and since again $X_2$ divides $X_2$
 $\hm(\hm(X_2 \rmult X_1) \rmult X_1) = X_2$.
But $\hm(X_2 \rmult X_1 \rmult X_1) = X_1$
In the next section we will show how using a restricted set of divisors only
 will enable transitivity.

We now want to outline possible generalizations of
 Gr\"obner bases to function rings.
We start with the more restricted case of function rings over fields, as this case
 is easier and more common in the literature.
It allows an approach similar to the one using standard representations
 and a similar algebraic characterization of Gr\"obner bases.