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We consider the ideal magnetohydrodynamics (MHD) subjected to a strongmagnetic field along $x_1$ direction in three-dimensional thin domains$\Omega_\delta=\mathbb{R}^2\times(-\delta,\delta)$ with slip boundaryconditions. It is well-known that in this situation the system will generateAlfv\'en waves. Our results are summarized as follows:  (i).\, We construct the global solutions (Alfv\'en waves) to MHD in the thindomain $\Omega_\delta$ with $\delta>0$. In addition, the uniform energyestimates are obtained with respected to the parameter $\delta$.  (ii). We justify the asymptotics of the MHD equations from the thin domain$\Omega_\delta$ to the plane $\mathbb{R}^2$. More precisely, we prove that the3D Alfv\'en waves in $\Omega_\delta$ will converge to the Alfv\'en waves in$\mathbb{R}^2$ in the limit that $\delta$ goes to zero. This shows thatAlfv\'en waves propagating along the horizontal direction of the (3D) strip arestable and can be approximated by the (2D) Alfv\'en waves when $\delta$ issufficiently small. Moreover, the control of the (2D) Alfv\'en waves can beobtained from the control of (3D) Alfv\'en waves in the thin domain$\Omega_\delta$ with aid of the uniform bounds.  The proofs of main results rely on the design of the proper energy functionaland the null structures of the nonlinear terms. Here the null structures meanstwo aspects: separation of the Alfv\'en waves ($z_+$ and $z_-$) and no badquadratic terms $Q(\partial_3 z_-^h, \partial_3 z_+^h)$ where $z_\pm=(z_\pm^h,z^3_\pm)$ and $Q(\partial_3 z_-^h,\partial_3 z_+^h)$ is the linear combinationof terms $\partial^\alpha \partial_3 z_-^h\partial^\beta \partial_3 z_+^h$ with$\alpha, \beta \in (\mathbb{Z}_{\geq0})^3$.

On the ideal magnetohydrodynamics in three-dimensional thin domains: well-posedness and asymptotics