We　present　the　improved　three-dimensional　axially　symmetric　incompressible　magnetohydrodynamics　(MHD)　equations　with　nonzero　swirl.　We　consider　three　kinds　of　smooth　axially　symmetric　particular　solutions　to　the　MHD　equations:　(1)　u=0,B-r=B-z=0,　(2)　B-r=B-z=0,　and　(3)　B=0.　In　particular,　we　derive　new　regularity　criteria　for　these　three　kinds　of　the　three-dimensional　axially　symmetric　smooth　solutions　to　the　MHD　equations.　Our　results　also　reveal　some　interesting　dynamic　behavior　of　the　interaction　by　the　angular　vorticity　field　and　the　angular　current　density　field　j.　Copyright　(c)　2016　John　Wiley　&　Sons,　Ltd.