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|　＜　mC,　we　show　that　the　nonlinear　RT　instability　still　occurs　by　developing　a　new　analysis　technique　based　on　the　method　of　bootstrap　instability.　The　current　result　reveals　from　the　mathematical　point　of　view　that　the　sufficiently　large　vertical　equilibrium　magnetic　field　has　a　stabilizing　effect　and　can　prevent　the　RT　instability　in　MHD　flows　from　occurring.　Also,　the　nonslip　boundary　condition　in　the　direction　of　the　equilibrium　magnetic　field　contributes　to　the　stabilizing　effect　of　magnetic　fields.　Similar　conclusions　can　also　be　verified　for　the　horizontal　magnetic　field　when　the　domain　is　vertically　periodic,　which　shows　that　the　horizontal　magnetic　field　has　the　same　stabilizing　effect　as　the　vertical　one.　©　2018　Society　for　Industrial　and　Applied　Mathematics.