  Whether the global existence and uniqueness of strong solutions of
$n$-dimensional incompressible magnetohydrodynamic (MHD for short) equations
with only kinematic viscosity or magnetic diffusion holds true or not remains
an outstanding open problem. In recent years, more attention has been paid to
the case when the magnetic field close to an equilibrium state (the background
magnetic field for short). Specifically, when the background magnetic field
satisfies the Diophantine condition (see (1.2) for details), Chen, Zhang and
Zhou [Sci. China Math. 41 (2022), pp.1-10] first studied the perturbation
system and established the decay estimates and stability of its solutions in 3D
periodic domain $\mathbb{T}^3$, which was then improved to
$H^{(3+2\beta)r+5+(\alpha+2\beta)}(\mathbb{T}^2)$ for 2D periodic domain
$\mathbb{T}^2$ and any $\alpha>0$, $\beta>0$ by Zhai [J. Differ. Equ. 374
(2023), pp.267-278]. In this paper, we seek to find the optimal decay estimates
and improve the space where the global stability is taking place. Through
deeply exploring and fully utilizing the structure of perturbation system, we
discover a new dissipative mechanism, which enables us to establish the decay
estimates in Sobolev space with much lower regularity. Based on the above
discovery, we greatly reduce the initial regularity requirement of
aforementioned two works from $H^{4r+7}(\mathbb{T}^3)$ and
$H^{(3+2\beta)r+5+(\alpha+2\beta)}(\mathbb{T}^2)$ to
$H^{(3r+3)^+}(\mathbb{T}^n)$ for $r>n-1$ when $n=3$ and $n=2$ respectively.
Additionally, we first present the linear stability result via the method of
spectral analysis in this paper. From which, the decay estimates obtained for
the nonlinear system can be seen as sharp in the sense that they are in line
with those for the linearized system.
