Modules with every subgenerated module lifting

It was shown in Dung-Smith [2] that, for a module M, every module in σ[M] is extending (CS module) if and only if every module in σ[M] is a direct sum of indecomposable modules of length 2 or, equivalently, every module in σ[M] is a direct sum of M-injective module and a semisimple module. Here we charcterize these modules by the fact that every module in σ[M] is lifting or, equivalently, decompose as a direct sum of a semisimple module and a projective module in σ[M]. They are also determined by the functor ring of σ[M] being a QF-2 ring with Jacobson radical square zero. As a corollary we obtain a result of Vanaja-Purav [8]: All (left) ^-modules are lifting if and only if R is a generalizad uniserial ring with Jacobson radical aquare zero.