Characterization of eventually periodic modules in the singularity categories

The singularity category of a ring makes only the modules finite projective dimension vanish among modules, so that is expected to characterize homological property infinite dimension. In this paper, such we deal with eventually periodic over left artin ring, and, as our main result, them in terms morphisms category. As applications, first prove that, for class dimensional algebras field, being preserved under singular equivalence Morita type level. Moreover, determine which connected Nakayama are when ground field algebraically closed.