Why a projective module is a projective cover for its largest semisimple quotient? That is  why the projection on the quotient is an essential morphism in this case?

Answer: if Q is a submodule of a projective module P which projects surjectively on the largest semisimple quotient of P, then Q projects surjectively on each simple quotient of P, and hence Q lies outside of any maximal submodule of P  contradiction. 

