  In the classical theory of formal languages, finite state automata allow to
recognize the words of a rational subset of $\Sigma^*$ where $\Sigma$ is a set
of symbols (or the alphabet). Now, given a semiring $(\K,+,.)$, one can
construct $\K$-subsets of $\Sigma^*$ in the sense of Eilenberg, that are
alternatively called noncommutative formal power series for which a framework
very similar to language theory has been constructed Particular noncommutative
formal power series, which are called rational series, are the behaviour of a
family of weighted automata (or $\K$-automata). In order to get an efficient
encoding, it may be interesting to point out one of them with the smallest
number of states. Minimization processes of $\K$-automata already exist for
$\K$ being: {\bf a)} a field, {\bf b)} a noncommutative field, {\bf c)} a PID .
When $\K$ is the bolean semiring, such a minimization process (with
isomorphisms of minimal objects) is known within the category of deterministic
automata. Minimal automata have been proved to be isomorphic in cases {\bf (a)}
and {\bf (b)}. But the proof given for (b) is not constructive. In fact, it
lays on the existence of a basis for a submodule of $\K^n$. Here we give an
independent algorithm which reproves this fact and an example of a pair of
nonisomorphic minimal automata. Moreover, we examine the possibility of
extending {\bf (c)}. To this end, we provide an {\em Effective Minimization
Process} (or {\em EMP}) which can be used for more general sets of
coefficients.
