For a given we call

**(3.2)**
The test for whether a module *M* will split is based on
the observation that a presentation matrix of *M* in block diagonal form
*Q*_{0} is
*J*_{0}- and *J*'_{0}-row-minimal (cf. Lemma 2).
It is possible to check by a standard basis computation
whether a row-minimal matrix *Q* is equivalent
to a block diagonal matrix.
To do that, we have to fix an arbitrary (local) monomial order
of *R*^{r}, ordering first by components; that is,
for any two monomials
and
the *i*-th unit
column.

**(3.3)**
Let
be a matrix formed by an ordered standard basis of
.
From the above order we obtain an integer *l*
such that all
columns of *Q*' having index *l*'> *l* will have their first
non-zero entry in the *k*'-th component, *k*'>*k*. Hence *Q*' will
have the block structure

and the modules associated to

Changing the roles of

Comparing these two different standard bases of

With the notation from above

**(3.4)**
Usually we are not in this situation and we first have to apply row operations.
But, assuming *J*_{0}-row-minimality, it is enough to apply only those
row-operations belonging to :

Letting , , we obtain

. Because of the minimality assumption,

Again, by Lemma 1

Consequently, we obtain

and, by the initial remark,
.

*(i2)* An easy computation shows that

Thus, , which shows that . On the other hand implies the existence of a such that . Because , there is an such that ; that is, and .