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Consider rectangular matrices over a commutative ring R. Assume the ideal ofmaximal minors factorizes, I_m(A)=J_1*J_2. When is A left-right equivalent to ablock-diagonal matrix? (When does the module/sheaf Coker(A) decompose as thecorresponding direct sum?) If R is not a principal ideal ring (or a closerelative of a PIR) one needs additional assumptions on A. No necessary andsufficient 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. Forexample, the passage to Noetherian/local/complete rings, the decomposability ofA over a graded ring R vs the decomposability of Coker(A) locally at the pointsof Proj(R), the restriction to a subscheme in Spec(R).  * The necessary and sufficient condition for decomposability of squarematrices in the case: det(A)=f_1*f_2 is not a zero divisor and f_1,f_2 areco-prime.  As an immediate application we give criteria of simultaneous (block-)diagonalreduction for tuples of matrices over a field, i.e. linear determinantalrepresentations.

Block-diagonal reduction of matrices over commutative rings I. (Decomposition of modules vs decomposition of their support)