Let R be a commutative Noetherian local ring with residue field k. We
show that if a finite direct sum of syzygy modules of k surjects onto `a
semidualizing module' or `a non-zero maximal Cohen-Macaulay module of finite
injective dimension', then R is regular. We also prove that R is regular if
and only if some syzygy module of k has a non-zero direct summand of finite
injective dimension.Comment: 7 page