Thomson's theorem implies that on the Bergman space over the unit disk if $h$ is holomorphic on the closed unit disk, then there is a finite Blaschke product $B$ such that $h$ can be written as a function of $B$, and the commutant of the multiplication operator $M_h$ by $h$ equals that of $M_B$. In this talk we will see that it is essentially generalized to the Bergman space over an annulus under a mild condition. The situation is complicated compared with the classical Bergman space over the unit disk. We also consider the associated reducing subspaces of concerned multiplication operator.