  Let $ R $ be a Cohen-Macaulay local ring. We prove that the $ n $th syzygy
module of a maximal Cohen-Macaulay $ R $-module cannot have a semidualizing
direct summand for every $ n \ge 1 $. In particular, it follows that $ R $ is
Gorenstein if and only if some syzygy of a canonical module of $ R $ has a
non-zero free direct summand. We also give a number of necessary and sufficient
conditions for a Cohen-Macaulay local ring of minimal multiplicity to be
regular or Gorenstein. These criteria are based on vanishing of certain Exts or
Tors involving syzygy modules of the residue field.
