A note on regular morphisms

Abstract. In this paper, we give some other characterizations of regular \(Hom(M,N)\). A property regular homomorphism via lifting is studied. Moreover, we also study regular submodules conditions and this is one of way to check regularity of \(Hom(M,N)\). On the other hand, we consider regular \(Hom(M,N)\) via weakly \(M\)-torsionless, idempotent submodules conditions. Finally, we give some results of \(\Delta\)-regular, \(\nabla\)-regular homomorphisms and some well-known results are obtained.