Abstract: In the paper, we define(inco) project modules of relatively hereditary torsion theory τ by intersection complement of module and study their properties; secondly, we define the(inco) τ-semisimple ring by(inco) τ-projective module and study their properties. When τ is a trivial torsion theory on R-mod, we prove that R is a semisimple ring if and only if R is a(inco) semisimple ring and satisfies(inco) condition.

Key words: Torsion theory τ , (inco) condition, τ -torsionfree module, (inco) τ -projective
module, semisimple ring