We　investigate　the　stabilizing　effect　of　the　vertical　equilibrium　magnetic　field　in　the　Rayleigh–Taylor　(RT)　problem　for　a　nonhomogeneous　incompressible　viscous　magnetohydrodynamic　(MHD)　fluid　of　zero　resistivity　in　the　presence　of　a　uniform　gravitational　field　in　a　horizontally　periodic　domain,　in　which　the　velocity　of　the　fluid　is　nonslip　on　both　upper　and　lower　flat　boundaries.　When　an　initial　perturbation　around　a　magnetic　RT　equilibrium　state　satisfies　some　relations,　and　the　strength　|m|　of　the　vertical　magnetic　field　of　the　equilibrium　state　is　bigger　than　the　critical　number　mC,　we　can　use　the　Bogovskii　function　in　the　standing-wave　form　and　adapt　a　two-tier　energy　method　in　Lagrangian　coordinates　to　show　the　existence　of　a　unique　global-in-time　(perturbed)　stability　solution　to　the　magnetic　RT　problem.　For　the　case　of　|m|　©　2018　Society　for　Industrial　and　Applied　Mathematics.