This is indeed a special case. The MU-homology of the sphere spectrum is the Lazard ring and hence is torsion-free, whereas all positive-degree elements of the stable homotopy ring are torsion by the Serre finiteness theorem and therefore belong to the aforementioned kernel.

Douglas Ravenel, Section 10.1 of: Localization with Respect to Certain Periodic Homology Theories, American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 351-414 (doi:10.2307/2374308, jstor:2374308)

Michael Hopkins, Jeffrey Smith, Nilpotence and Stable Homotopy Theory II, Annals of Mathematics Second Series, Vol. 148, No. 1 (Jul., 1998), pp. 1-49 (jstor:120991)