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**(5.1)**
Fix
Consider all those matrices
from
which do not increase
,
and set

Because
and
if and only if
,
we obtain:

Choose a basis
of the module
and
a set of associated matrices
.

**Proposition 9** ((sufficient condition for

*J*-row-minimality))

*Q* is

*J*-row-minimal if either

(i1)
do not have a unit entry; i.e.,

or

(i2) for any
,
the corresponding
matrix
satisfies the condition:
is nilpotent.

Condition *(i1)* implies *(i2)*. Assuming *(i2)*, we obtain for any

because
is invertible. Therefore *Q* is *J*-row-minimal.

**(5.2)**
In order to obtain a necessary condition for row-minimality, we have to improve
the choice of a good representative
in the equivalence class of *Q*, cf. (3.3).
Write
in block form

We may assume that the columns of *A* form a completely reduced, minimal
basis of
.
This means the columns of *A* form a minimal system of
generators, increasingly ordered, and the *l*-th generator is reduced with
respect to the standard basis generated by the first *l*-1 generators
for appropriate *l*.

**Lemma 10**
Let

*Q* be a submodule of

*R*^{k}, given by a completely reduced, minimal
basis

.
A submodule

*P* of

*Q* is proper
if and only if

*lead*(

*P*),
the initial module, does not contain all leading terms of

*A*.

We want to show that for a proper submodule *P* the minimal leading
term not contained in *lead*(*Q*) must be in *lead*(*A*). Assume the contrary; i.e.
the minimal
is not contained in *lead*(*A*).
We introduce the module
,
generated by all monomials
in *R*^{k} which are greater than
.
Considering the whole situation in the factor module *R*^{n}/*S*, we have
the equation

and, from the assumption on reducedness, it follows that all *g*_{i}
belong to the maximal ideal
.
The staircases of both modules *Q* and *P* are identical up to
,
and,
hence, for every generator
an equation

with
exists. We obtain

The second summand vanishes in *R*^{n}/*S*. Thus,
must be a leading term of *P*,
too.
The other implication is a well-known fact about standard basis.

**(5.3)**
Without loss of generality we will assume that
fulfills the conditions of (5.2).
For
we obtain
and
Therefore, matrices

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** Up:** Splitting algorithm for vector
** Previous:** 4. Splitting criteria
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