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**(4.1)**
Next we want to characterize those row-minimal matrices belonging to a
decomposable module. The idea comes from the fact that the canonical homomorphism
factors via *M*
and that the induced surjection
is an isomorphism. Hence, it reflects a direct sum decomposition
of *M* if and only if *Q* is equivalent to a block diagonal matrix *Q*_{0}. Then *Q* is necessarily row-minimal with respect to *J*_{0} and *J*'_{0}.

**Proposition 6**
If

*Q* is row-minimal, then

is decomposable if and only if for some
permutation

.

Assume *M* splits. Then we know
and,
for some permutation ,
;
i.e.,
and
By Proposition 5*(i2)*, we obtain

The same holds for *J*'_{0} , because *Q* is *J*'_{0}-row-minimal, too; that is,

But then the composition of the canonical surjection

with

must be an isomorphism, showing
.

Note:
and
are computed by a standard
basis computation with respect to a certain module ordering in the components.
So we simply have to check whether

has a solution *X*, which corresponds to a simple syzygy computation.

**(4.2)**
Before computing a solution *X*, it is necessary for *Q* to be row-minimal,
which may be tested using methods discussed in the next section. Usually
*Q* is not row-minimal, in which case we shall use an algorithm to
transform *Q* into an equivalent and *J*_{0}-row-minimal matrix.
But we do not know a procedure
to obtain row-minimality for all subsets immediately.
Hence we shall formulate a second version of
the splitting criteria.

**Proposition 7**
If

*Q* is

*J*_{0}-row-minimal, then

By Proposition 5, it follows that

then
and
.
Therefore

and
,
giving

from which it may be concluded that

The other direction is obvious.
Note:

if and only if
*C*' - *VA*' = *D*'*V*'
for some *V*'; i.e. we obtain

**Corollary 8**
If

*Q* is

*J*_{0}-row-minimal,
then for an ordered standard basis

of

*Q* we have:

if and only if

the equation

*VA*'+

*D*'

*V*'=

*C*' has a solution (

*V*,

*V*').

Deciding the solvability of the above equation is reduced to
a lifting computation: Let
and
be matrices corresponding
to
and
,
and let [*V*], [*V*'], [*C*']
denote flattenings to a column vector of
*V*, *V*', *C*', respectively. Then we obtain

that is, [*C*'] can be lifted to
if and only if the
equation has a solution.

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** Previous:** 3. Row-minimal matrices
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