LetRbe a ring graded by an abelian group.We study prime ideals ofRthat are maximal for not containing nonzero homogeneous elements.Also prime ideals of the symmetric graded Martindale ring of quotients ofRare investigated.The results are applied to study whenRis a Jacobson ring in caseRis aZ-graded ring or a group ring of a finitely generated abelian group, or in caseRis right Noetherian and strongly graded by a polycyclic-by-finite group.