  Let M be a matrix whose entries are power series in several variables and
determinant det(M) does not vanish identically. The equation det(M)=0 defines a
hypersurface singularity and the (co)-kernel of M is a maximally Cohen-Macaulay
module over the local ring of this singularity.
  Suppose the determinant det(M) is reducible, i.e. the hypersurface is locally
reducible. A natural question is whether the matrix is equivalent to a
block-diagonal or at least to an upper-block-triangular. (Or whether the
corresponding module is decomposable or at least is an extension.) We give
various necessary and sufficient criteria.
  Two classes of such matrices of functions appear naturally in the study of
decomposability: those with many generators (e.g. maximally generated or Ulrich
maximal) and those that descend from birational modifications of the
hypersurface by pushforwards (i.e. correspond to modules over bigger rings).
Their properties are studied.
