  In this paper, we study a system of stochastic partial differential equations
with slow and fast time-scales, where the slow component is a stochastic real
Ginzburg-Landau equation and the fast component is a stochastic
reaction-diffusion equation, the system is driven by $\alpha$-stable process
with $\alpha\in (1,2)$. Using the classical Khasminskii approach based on time
discretization and the techniques of stopping times, we show that the slow
component strong converges to the solution of the corresponding averaged
equation under some suitable conditions.
