It is known that for a module with projective cover, its direct summands need not have a projective cover. Let \(p\colon P\to M\) be a projective cover of \(_RM\), with \(K=\ker p\), \(E=\text{End}_R(P)\), \(S=\{\alpha\in E:K\alpha\subseteq K\}\) and \(T=\{\alpha\in E:P\alpha\subseteq K\}\). It is shown that every direct summand of \(M\) has a projective cover if and only if idempotents lift modulo \(T\) in \(S\). This is also equivalent to each direct summand of the left \(S\)-module \(S/T\) having a projective cover.
Liftings of idempotents are studied further because of their relationship with the existence of projective covers. Let \(L\) be an ideal of the ring \(R\) and let \(I(L)=\{r\in R\mid Lr\subseteq L\}\). It is proved that the following conditions are equivalent for a ring \(R\): (a) Idempotents lift modulo \(J(R)\) in \(R\); (b) Idempotents lift modulo \(I(L)\); (c) For any ideal \(L\subseteq J(R)\) every direct summand of the left \(R\)-module \(R/L\) has a projective cover; (d) Every direct summand of a cyclic left \(R\)-module, with projective cover, has a projective cover.
Every direct summand of a finitely generated left \(R\)-module, with projective cover, is shown to have a projective cover if and only if idempotents lift modulo \(M_n(J)\) in \(M_n(R)\) for every natural number \(n\). A few more sufficient conditions for direct summands to inherit projective covers are found. The paper concludes with an open question and a few applications.