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This paper examines the initial-value problem for the two-dimensional magnetohydrodynamic equation with only magnetic diffusion (without velocity dissipation). Whether or not its classical solutions develop finite time singularities is a difficult problem and remains open. This paper establishes two main results. The first result features a regularity criterion in terms of the magnetic field. This criterion comes naturally from our approach to obtain a global bound for the vorticity. Due to the lack of velocity dissipation, it is difficult to conclude the boundedness of the vorticity from the vorticity equation itself. Instead we derive and involve a new equation for the combined quantity of the vorticity and a singular integral operator on the tensor product of the magnetic field. This criterion may be verifiable. Our second main result is a weaker version of the small data global existence result, which is shown by the bootstrap argument.