We introduce and study the notion of w-Rickart modules (i.e. modules M such that for every nonzero endomorphism phi of M, the kernel of phi is contained in a proper direct summand of M). We show that the class of right w-Rickart modules lies properly between the class of right K-nonsingular modules and the class of right Rickart modules. Many properties of w-Rickart modules are established. Some relevant counterexamples are indicated.