  Consider rectangular matrices over a commutative ring R. Assume the ideal of
maximal minors factorizes, I_m(A)=J_1*J_2. When is A left-right equivalent to a
block-diagonal matrix? (When does the module/sheaf Coker(A) decompose as the
corresponding direct sum?) If R is not a principal ideal ring (or a close
relative of a PIR) one needs additional assumptions on A. No necessary and
sufficient criterion for such block-diagonal reduction is known.
  In this part we establish the following:
  * The persistence of (in)decomposability under the change of rings. For
example, the passage to Noetherian/local/complete rings, the decomposability of
A over a graded ring R vs the decomposability of Coker(A) locally at the points
of Proj(R), the restriction to a subscheme in Spec(R).
  * The necessary and sufficient condition for decomposability of square
matrices in the case: det(A)=f_1*f_2 is not a zero divisor and f_1,f_2 are
co-prime.
  As an immediate application we give criteria of simultaneous (block-)diagonal
reduction for tuples of matrices over a field, i.e. linear determinantal
representations.
